Fundamental mistuning model for determining system properties and predicting vibratory response of bladed disks

ABSTRACT

A reduced order model called the Fundamental Mistuning Model (FMM) accurately predicts vibratory response of a bladed disk system. The FMM software may describe the normal modes and natural frequencies of a mistuned bladed disk using only its tuned system frequencies and the frequency mistuning of each blade/disk sector (i.e., the sector frequencies). The FMM system identification methods—basic and advanced FMM ID methods—use the normal (i.e., mistuned) modes and natural frequencies of the mistuned bladed disk to determine sector frequencies as well as tuned system frequencies. FMM may predict how much the bladed disk will vibrate under the operating (rotating) conditions. Field calibration and testing of the blades may be performed using traveling wave analysis and FMM ID methods. The FMM model can be generated completely from experimental data. Because of FMM&#39;s simplicity, no special interfaces are required for FMM to be compatible with a finite element model. Because of the rules governing abstracts, this abstract should not be used to construe the claims.

REFERENCE TO RELATED APPLICATION

This application is a divisional of copending U.S. application Ser. No.10/836,422 filed Apr. 30, 2004 and entitled “Fundamental Mistuning Modelfor Determining System Properties and Predicting Vibratory Response ofBladed Disks”, which claims priority benefits of the earlier filed U.S.provisional patent application Ser. No. 60/474,083, titled “FundamentalMistuning Model for Determining System Properties and PredictingVibratory Response of Bladed Disks,” filed on May 29, 2003, the entiretyof which is hereby incorporated by reference.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH

The invention in the present application was made under a grant from theUnited States Air Force Research Laboratory, Contract No.F33615-01-C-2186. The United States federal government may have certainrights in the invention.

BACKGROUND

The present disclosure generally relates to identification of mistuningin rotating, bladed structures, and, more particularly, to thedevelopment and use of reduced order models as an aid to theidentification of mistuning.

It is noted at the outset that the term “bladed disk” is commonly usedto refer to any (blade-containing) rotating or non-rotating part of anengine or rotating apparatus without necessarily restricting the term torefer to just a disk-shaped rotating part. Thus, a bladed disk can haveexternally-attached or integrally-formed blades or any other suitablerotating protrusions. Also, this rotating mechanism may have anysuitable shape, whether in a disk form or not. Further, the term “bladeddisk” may include stators or vanes, which are non-rotating bladed disksused in gas turbines. Various types of devices such as fans, pumps,turbochargers, compressors, engines, turbines, and the like, may becommonly referred to as “rotating apparatus.”

FIG. 1 illustrates a bladed disk 10 which is representative of thoseused in gas turbine engines. One such exemplary gas turbine 12 isillustrated in FIG. 1. Bladed disks used in turbine engines arenominally designed to be cyclically symmetric. If this were the case,then all blades would respond with the same amplitude when excited by atraveling wave. However, in practice, the resonant amplitudes of theblades are very sensitive to small changes in their properties. Thesmall variations that result from the manufacturing process and wearcause some blades to have a significantly higher response and may causethem to fail from high cycle fatigue (HCF). This phenomenon is referredto as the mistuning problem, and has been studied extensively.

FIG. 2 represents an exemplary selection of nodal diameter modes 13-15in a bladed disk. In the zero nodal diameter mode 13 (part (a) in FIG.2), all the blades move in phase with one another, while in the highernodal diameter modes 14-15, the blades move out of phase. FIG. 2(b)illustrates a mode 14 with five (5) nodal diameters, whereas FIG. 2(c)illustrates a mode 15 with ten (10) nodal diameters. The displacement ofthe blades as a function of angular position is given in these modes byfunctions sin(nθ) and cos(nθ), where “n” defines the number of nodaldiameters. For a given value of “n”, the corresponding sine and cosinemodes both have the same natural frequency. The only nodal diametermodes which do not have repeated frequencies are the cases of n=0 andn=N/2, where N is the number of blades on the disk.

FIG. 3 is an exemplary nodal diameter map 16 of a bladed disk's naturalfrequencies. Thus, the natural frequencies of a bladed disk are plottedas a function of the number of nodal diameters in their correspondingmode. When plotted in this fashion, the frequencies cluster intofamilies of modes. Each family consists of a set of N nodal diametermodes. Each mode 17-19 at the right in FIG. 3 represents the bladedeformation in the corresponding family. Although the relativeamplitudes of the blades varies from one nodal diameter to the next, thedeformation within a given blade remains uniform throughout all modeswithin a given family, at least for families which are isolated infrequency from their neighbors. The blade deformation in the first fewfamilies generally resembles the simple bending and torsion modes of acantilevered plate. Many families of modes have most of their strainenergy stored in the blades. These families appear relatively flat inFIG. 3, because the added strain energy introduced with additional nodaldiameters has a minimal effect on their natural frequency. In contrast,mode families with large strain energy in the disk increase theirfrequency rapidly from one nodal diameter to the next.

Mistuning can significantly affect the vibratory response of a bladeddisk. This sensitivity stems from the nature of the eigenvalue problemthat describes a disk's modes and natural frequencies. An eigenvalue isequal to the square of the natural frequency of a mode. The eigenvaluesof a bladed disk are inherently closely spaced due to the system'srotationally periodic design, as can be seen from the plot in FIG. 3.Therefore, the eigenvectors (mode shapes) of a bladed disk are verysensitive to the small perturbations caused by mistuning. In the case ofvery small mistuning, the blade displacements in the modes are given bydistorted sine and cosine waves, while large mistuning can alter themodes to such an extent that the majority of the motion will belocalized to just one or two blades. FIG. 4 illustrates exemplary forcedresponse tracking plots 20-21 of a tuned bladed disk system (plot 20)and the mistuned disk system (plot 21). The plot 21 illustrates bladeamplitude magnification caused by mistuning.

To address the mistuning problem, researchers have developed reducedorder models (ROMs) of the bladed disk. These ROMs have the structuralfidelity of a finite element model of the full rotor, while incurringcomputational costs that are comparable to that of a mass-spring model.In numerical simulations, most published ROMs have correlated extremelywell with numerical benchmarks. However, some models have at times haddifficulty correlating with experimental data. These results suggestthat the source of the error may lie in the inability to determine thecorrect input parameters to the ROMs.

The standard method of measuring mistuning in rotors with attachableblades is to mount each blade in a broach block and measure its naturalfrequency. The difference of each blade's natural frequency from themean value is then taken as a measure of the mistuning. However, themistuning measured through this method may be significantly differentfrom the mistuning present once the blades are mounted on the disk. Thisvariation in mistuning can arise because each blade's frequency isdependent on the contact conditions at the attachment. Not only may theblade-broach contact differ from the blade-disk contact, but the contactconditions can also vary from slot-to-slot around the wheel or disk.Therefore, to accurately measure mistuning, it is desirable to developmethods that can infer the mistuning from the vibratory response of theblade-disk assembly as a whole. In addition, many blade-disk structuralsystems are now manufactured as a single piece in which the bladescannot be physically separated from the disk. In the gas turbineindustry they are referred to as blisks (for bladed disks) or IBRs (forintegrally bladed rotors). Thus, in the case of IBRs, the conventionaltesting methods of separately measuring individual blade frequenciescannot be applied and, therefore, it is desirable to develop methodsthat can infer the properties of the individual blades from the behaviorof the blade-disk assembly as a whole.

Therefore, to accurately measure mistuning, it is desirable to developmethods or reduced order models that apply to individual blades or theblade-disk assembly as a whole. It is further desirable to use themistuning values obtained from the newly-devised reduced order models toverify finite element models of the system and also to monitor thefrequencies of individual blades to determine if they have changedbecause of cracking, erosion or other structural changes. It is alsodesirable that the obtained mistuning values can be analyticallyadjusted and used with the reduced order model to predict the vibratoryresponse of the structure (or bladed disk) when it is in use, e.g., whenit is rotating in a gas turbine engine, an industrial turbine, a fan, orany other rotating apparatus.

SUMMARY

In one embodiment, the present disclosure contemplates a method thatcomprises obtaining a set of vibration measurements that providesfrequency deviation of each blade of a bladed disk system from the tunedfrequency value of the blade and nominal frequencies of the bladed disksystem when tuned; and calculating the mistuned modes and naturalfrequencies of the bladed disk system from the blade frequencydeviations and the nominal frequencies.

In another embodiment, the present disclosure contemplates a method thatcomprises obtaining nominal frequencies of a bladed disk system whentuned; measuring at least one mistuned mode and natural frequency of thebladed disk system; and calculating mistuning of at least one blade (orblade-disk sector) in the bladed disk system from only the nominalfrequencies and the at least one mistuned mode and natural frequency.

In a further embodiment, the present disclosure contemplates a methodthat comprises measuring a set of mistuned modes and natural frequenciesof a bladed disk system; and calculating mistuning of at least one bladein the bladed disk system and nominal frequencies of the bladed disksystem when tuned by using only the set of mistuned modes and naturalfrequencies.

In a still further embodiment, the present disclosure contemplates amethod that comprises obtaining frequency response data of each blade ina bladed disk system to a traveling wave excitation; transforming datarelated to spatial distribution of the traveling wave excitation and thefrequency response data; and determining a set of mistuned modes andnatural frequencies of the bladed disk system using data obtained fromthe transformation.

According to the methodology of the present disclosure, a reduced ordermodel called the Fundamental Mistuning Model (FMM) is developed toaccurately predict vibratory response of a bladed disk system. FMM maydescribe the normal modes and natural frequencies of a mistuned bladeddisk using only its tuned system frequencies and the frequency mistuningof each blade/disk sector (i.e., the sector frequencies). If the modaldamping and the order of the engine excitation are known, then FMM canbe used to calculate how much the vibratory response of the bladed diskwill increase because of mistuning when it is in use. The tuned systemfrequencies are the frequencies that each blade-disk and blade wouldhave were they manufactured exactly the same as the nominal designspecified in the engineering drawings. The sector frequenciesdistinguish blade-disks with high vibratory response from those with alow response. The FMM identification methods—basic and advanced FMM IDmethods-use the normal (i.e., mistuned) modes and natural frequencies ofthe mistuned bladed disk measured in the laboratory to determine sectorfrequencies as well as tuned system frequencies. Thus, one use of theFMM methodology is to: identify the mistuning when the bladed disk is atrest, to extrapolate the mistuning to engine operating conditions, andto predict how much the bladed disk will vibrate under the operating(rotating) conditions.

In one embodiment, the normal modes and natural frequencies of themistuned bladed disk are directly determined from the disk's vibratoryresponse to a traveling wave excitation in the engine. These modes andnatural frequency may then be input to the FMM ID methodology to monitorthe sector frequencies when the bladed disk is actually rotating in theengine. The frequency of a disk sector may change if the blade'sgeometry changes because of cracking, erosion, or impact with a foreignobject (e.g., a bird). Thus, field calibration and testing of the blades(e.g., to assess damage from vibrations in the engine) may be performedusing traveling wave analysis and FMM ID methods together.

The FMM software (containing FMM ID methods) may receive the requisiteinput data and, in turn, predict bladed disk system's mistuning andvibratory response. Because the FMM model can be generated completelyfrom experimental data (e.g., using the advanced FMM ID method), thetuned system frequencies from advanced FMM ID may be used to validatethe tuned system finite element model used by industry. Further, FMM andFMM ID methods are simple, i.e., no finite element mass or stiffnessmatrices are required. Consequently, no special interfaces are requiredfor FMM to be compatible with a finite element model.

BRIEF DESCRIPTION OF THE DRAWINGS

For the present disclosure to be easily understood and readilypracticed, the present disclosure will now be described for purposes ofillustration and not limitation, in connection with the followingfigures, wherein:

FIG. 1 illustrates a bladed disk which is representative of those usedin gas turbine engines;

FIG. 2 represents an exemplary selection of nodal diameter modes in abladed disk;

FIG. 3 is an exemplary nodal diameter map of a bladed disk's naturalfrequencies;

FIG. 4 illustrates exemplary forced response tracking plots of a tunedbladed disk system and the mistuned disk system;

FIG. 5 illustrates near equivalence of sector modes from various nodaldiameters;

FIG. 6 illustrates an exemplary three dimensional (3D) finite elementmodel of a high pressure turbine (HPT) blade-disk sector;

FIG. 7 shows tuned system frequencies of the first families of modes ofthe blade-disk sector modeled in FIG. 6;

FIG. 8 illustrates the tuned frequencies of the fundamental family ofmodes of the system modeled in FIG. 6, along with the frequencies asdetermined by ANSYS® software and the best-fit mass-spring model;

FIGS. 9 (a) and (b) depict representative results of using FMM with arealistic mistuned bladed disk modeled in FIG. 6;

FIGS. 10(a) and (b) show a representative case of the blade amplitudesas a function of excitation frequency for a 7^(th) engine orderexcitation predicted by the mass-spring model, ANSYS® software, and FMM;

FIG. 11 illustrates the leading edge blade tip displacements for thethird family of modes shown in FIG. 7;

FIGS. 12(a)-(c) illustrate FMM and ANSYS® software predictions of bladeamplitude as a function of excitation frequency for a 2^(nd) engineorder excitation of 2^(nd), 3^(rd), and 4^(th) families respectively;

FIGS. 13(a)-(c) illustrate FMM and ANSYS® software predictions of bladeamplitude as a function of excitation frequency for a 7^(th) engineorder excitation of 2^(nd), 3^(rd), and 4^(th) families respectively;

FIG. 14 represents an exemplary finite element model of a twenty bladecompressor;

FIG. 15 illustrates the natural frequencies of the tuned compressormodeled in FIG. 14;

FIG. 16 shows the comparison between the sector mistuning calculateddirectly by finite element simulations of each mistuned blade/sector andthe mistuning identified by basic FMM ID;

FIG. 17 schematically illustrates a rotor 29 with exaggerated staggerangle variations as viewed from above;

FIG. 18 shows a representative mistuned mode caused by stagger anglemistuning of the rotor in FIG. 14;

FIG. 19 illustrates a comparison of mistuning determination from basicFMM ID and the variations in the stagger angles;

FIG. 20 depicts a comparison of mistuning predicted using advanced FMMID with that obtained using the finite element analysis (FEA);

FIG. 21 shows a comparison of the tuned frequencies identified byadvanced FMM ID and those computed directly with the finite elementmodel;

FIG. 22 illustrates an exemplary setup to measure transfer functions oftest rotors and also to verify various FMM ID methods;

FIG. 23 illustrates natural frequencies of a test compressor with nomistuning;

FIG. 24 illustrates a typical transfer function from a test compressorobtained using the test setup shown in FIG. 22;

FIG. 25 illustrates a comparison of mistuning from each FMM ID methodwith benchmark results for a test rotor SN-1;

FIG. 26 shows a comparison of tuned system frequencies for the testrotor SN-1 from advanced FMM ID and the finite element model (FMM) usingANSYS® software;

FIGS. 27 and 28 are similar to FIGS. 25 and 26, respectively, butillustrate the identified mistuning and tuned system frequencies for adifferent test rotor SN-3;

FIGS. 29 (a) and (b) show a comparison, for rotors SN-1 and SN-3respectively, of the mistuning identified by FMM ID with the values frombenchmark results from geometric measurements;

FIG. 30 illustrates a comparison of tuned system frequencies fromadvanced FMM ID and ANSYS® software for torsion modes of rotors SN-1 andSN-3;

FIG. 31(a) depicts FMM-based forced response data, whereas FIG. 31(b)depicts the experimental forced response data;

FIGS. 32(a) and (b) respectively show relative blade amplitudes atforced response resonance for the resonant peaks labeled {circle around(1)} and {circle around (2)} in FIG. 31(a);

FIG. 33 depicts cumulative probability function plots of peak bladeamplitude for a nominally tuned and nominally mistuned compressor;

FIG. 34 shows mean and standard deviations of each sector's mistuningfor a nominally mistuned compressor;

FIG. 35 illustrates a lumped parameter model of a rotating blade;

FIG. 36 shows a comparison of mistuning values analytically extrapolatedto speed with an FEA (finite element analysis) benchmark;

FIG. 37 illustrates the effect of centrifugal stiffening on tuned systemfrequencies;

FIG. 38 illustrates the effect of centrifugal stiffening on mistuning;

FIG. 39 depicts frequency response of blades to a six engine orderexcitation at 40,000 RPM rotational speed;

FIGS. 40(a), (b) and (c) show a comparison of the representative modeshape extracted from the traveling wave response data with benchmarkresults for a stationary benchmark;

FIG. 41 depicts comparison of the natural frequencies extracted from thetraveling wave response data with the benchmark results for thestationary benchmark of FIG. 40;

FIG. 42 shows a calibration curve relating the effect of a unit mass ona sector's frequency deviation in a stationary benchmark;

FIG. 43 shows the comparison between the mass mistuning identifiedthrough traveling wave FMM ID with the values of the actual massesplaced on each blade tip;

FIGS. 44(a) and (b) show tracking plots of blade amplitudes as afunction of excitation frequency for two different acceleration rates;

FIGS. 45(a) and (b) illustrate the comparison of the mistuningdetermined through the traveling wave system identification method withbenchmark values for two different acceleration rates; and

FIG. 46 illustrates an exemplary process flow depicting various bladesector mistuning tools discussed herein.

DETAILED DESCRIPTION

Reference will now be made in detail to some embodiments of the presentdisclosure, examples of which are illustrated in the accompanyingdrawings. It is to be understood that the figures and descriptions ofthe present disclosure included herein illustrate and describe elementsthat are of particular relevance to the present disclosure, whileeliminating, for the sake of clarity, other elements found in typicalbladed disk systems, engines, or rotating devices. It is noted here thatthe although the discussion given below is primarily with reference to ablade-disk sector, the principles given below can be equally applied tojust the blade portion of the blade-disk sector as can be appreciated byone skilled in the art. Therefore, the terms “blade” and “blade-disksector” have been used interchangeably in the discussion below, and noadditional discussion of the blade-only application is presented herein.

[1] DERIVING THE FUNDAMENTAL MISTUNING MODEL (FMM)

The more general form of the modal equation for the FundamentalMistuning Model (FMM), derived below, is applicable to rotating, bladedapparatus. The generalized FMM formulation differs in two ways from theoriginal FMM derivation described in the paper by D. M. Feiner and J. H.Griffin titled “A Fundamental Model of Mistuning for a Single Family ofModes,” appearing in the Proceedings of IGTI, ASME Turbo Expo 2002 (Jun.3-6, 2002), Amsterdam, The Netherlands. This paper is incorporatedherein by reference in its entirety. First, the following derivation nolonger approximates the tuned system frequencies by their average value.This allows for a much larger variation among the tuned frequencies.Second, rather than using the blade-alone mode as an approximation ofthe various nodal diameter sector modes, a representative mode of asingle blade-disk sector is used below. Consequently, the approach nowincludes the disk portion of the mode shape, and thus allows for morestrain energy in the disk. Although mistuning may be measured as apercent deviation in the blade-alone frequency (as in the abovementioned paper), in the following discussion mistuning is measured as apercent deviation in the frequency of each blade-disk sector. The sectorfrequency deviations not only capture mistuning in the blade, but canalso capture mistuning in the disk as well as variations in the ways theblades are attached to the disk.

In the discussion below, section 1.1 describes how the subset of nominalmodes (SNM) approach is used to reduce the order of the mistunedfree-response equations and formulates the problem in terms of reducedorder sector matrices, section 1.2 relates the sector matrices tomistuned sector frequencies, and section 1.3 simplifies the resultingmathematical expressions.

1.1 Reduction of Order

Consider a mistuned, bladed disk in the absence of an excitation. Theorder of its equation of motion may be reduced through a subset ofnominal modes (SNM) approach. The resulting reduced order equation canbe written as (see, for example, the discussion in the above mentionedFeiner-Griffin paper):[(Ω°² +Δ{circumflex over (K)})−ω_(j) ²(I+Δ{circumflex over (M)})]{rightarrow over (β)}_(j)=0  (1)Ω°² is a diagonal matrix of the tuned system eigenvalues (an eigenvalueis equal to the square of the natural frequency of a mode), and I is theidentity matrix. Δ{circumflex over (K)} and Δ{circumflex over (M)} arethe variations in the modal stiffness and modal mass matrices caused bystiffness and mass mistuning. The vector {right arrow over (β)}_(j),contains weighting factors that describe the j^(th) mistuned mode as alimited sum of tuned modes, i.e.,{right arrow over (φ)}_(j)=Φ°{right arrow over (β)}_(j)  (2)where Φ° is a matrix whose columns are the tuned system modes.

Note that to first order, (I+Δ{circumflex over (M)})⁻¹≈(I−Δ{circumflexover (M)}). Thus by pre-multiplying (1) by (I+Δ{circumflex over (M)})⁻¹and keeping only first order terms, the expression becomes(Λ°+Â) {right arrow over (β)}_(j)=ω_(j) ²{right arrow over (β)}_(j)  (3)whereÂ=Δ{circumflex over (K)}−Δ{circumflex over (M)}Ω° ²  (4)

The next section relates the matrix Â to the frequency deviations of themistuned sectors.

1.2 Relating Mistuning to Sector Frequency Deviations

Relating Â to frequency deviations of the sectors is a three-stepprocess. First, the mistuning matrix is express in terms of the systemmode shapes of an individual sector. Then, the system sector modes arerelated to the corresponding mode of a single, isolated sector. Finally,the resulting sector-mode terms in Â are expressed in terms of thefrequency deviations of the sectors.

1.2.1 Relating Mistuning to System Sector Modes

Consider the mistuning matrix, Â, in equation (4). This matrix can beexpressed as a sum of the contributions from each mistuned sector.$\begin{matrix}{\hat{A} = {\sum\limits_{s = 0}^{N - 1}{\hat{A}}^{(s)}}} & (5)\end{matrix}$where the superscript denotes that the mistuning corresponds to thes^(th) sector. The expression for a single element of Â^((s)) isÂ _(mn) ^((s))={right arrow over (φ)}°_(m) ^((s)H)(ΔK ^((s))−ω°_(n) ² ΔM^((s))) {right arrow over (φ)}°_(n) ^((s))  (6)where ΔK^((s)) and ΔM^((s)) are the physical stiffness and massperturbations of the s^(th) sector. The modes {right arrow over(φ)}°_(m) ^((s)) and {right arrow over (φ)}°_(n) ^((s)); are theportions of the m^(th) and n^(th) columns of Φ° which describe thes^(th) sector's motion. The term ω°_(n) ² is the nth diagonal element ofΩ°². Equation (6) relates the mistuning to the system sector modes. Inthe next section, these modes are related to the mode of a singleisolated blade-disk sector.

1.2.2 Relating System Sector Modes to an Average Sector Mode

The tuned modes in equation (6) are expressed in a complex travelingwave form. Thus, the motion of the s^(th) sector can be related to themotion of the 0^(th) sector by a phase shift. Thus, equation (6) can berestated as $\begin{matrix}{{\hat{A}}_{mn}^{(s)}{\mathbb{e}}^{{\mathbb{i}}\quad{s{({n - m})}}\frac{2\pi}{N}}{{\overset{->}{\phi}}_{m}^{{\bullet{(0)}}H}\left( {{\Delta\quad K^{(s)}} - {\omega_{n}^{\circ 2}\Delta\quad M^{(s)}}} \right)}{\overset{->}{\phi}}_{n}^{\bullet{(0)}}} & (7)\end{matrix}$Because the tuned modes used in the SNM formulation are an isolatedfamily of modes, the sector modes of all nodal diameters look nearlyidentical as can be seen from FIG. 5, which illustrates near equivalenceof sector modes from various nodal diameters. Therefore, one canapproximate the various sector modes by an average sector mode. Applyingthe average sector mode approximation for the system sector modes inequation (7), Â_(mn) ^((s)) can be written as $\begin{matrix}{{\hat{A}}_{mn}^{(s)} = {\left( \frac{\omega_{m}^{*}\omega_{n}^{*}}{\omega_{V}^{\bullet 2}} \right){{\mathbb{e}}^{{\mathbb{i}}\quad{s{({n - m})}}\frac{2\pi}{N}}\left\lbrack {{{\overset{->}{\psi}}^{{\bullet{(0)}}^{H}}\left( {{\Delta\quad K^{(s)}} - {\omega_{n}^{\bullet 2}\Delta\quad M^{(s)}}} \right)}{\overset{->}{\psi}}^{\bullet{(0)}}} \right\rbrack}}} & (8)\end{matrix}$where {right arrow over (ψ)}°⁽⁰⁾ is the average tuned system sectormode, and ω°_(ψ) is its natural frequency. In practice, {right arrowover (ψ)}°⁽⁰⁾ can be taken to be the median modal diameter mode. Thefactor (ω°_(m)ω°_(n))/(ω°_(ψ) ²) scales the average sector mode terms sothat they have approximately the same strain energy as the sector modesthey replace.

1.2.3 Introduction of Sector Frequency Deviation

The deviation in a sector frequency quantity may be used to measuremistuning. To understand this concept, consider an imaginary “test”rotor. In the test rotor every sector is mistuned in the same fashion,so as to match the mistuning in the sector of interest. Since the testrotor's mistuning is cyclically symmetric, its mode shapes are virtuallyidentical to those of the tuned system. However, there will be a shiftin the tuned system frequencies. For small levels of mistuning, thefrequency shift is nearly the same in all of the tuned system modes andcan be approximated by the fractional change in the frequency of themedian nodal diameter mode. This may typically be the case for anisolated family of modes in which the strain energy is primarily in theblades. If there is a significant amount of strain energy in the diskthen the frequency of the modes may change significantly as a functionof nodal diameter and the modes may not be isolated (i.e., the modes maycover such a broad frequency range that they may interact with otherfamilies of modes). However, in the following, the fractional shift inthe median nodal diameter's frequency is taken as a measure of mistuningand is defined as the sector frequency deviation.

The bracketed terms of in equation (8) are related to these frequencydeviations in the following manner. Consider a bladed disk that ismistuned in a cyclic symmetric fashion, i.e., each sector undergoes thesame mistuning. Its free-response equation of motion is given by theexpression[(K°+ΔK)−ω_(n) ²(M°+ΔM)]{right arrow over (φ)}_(n)=0  (9)Take the mode {right arrow over (φ)}_(n) to be the mistuned version ofthe tuned median nodal diameter mode, {right arrow over (ψ)}°. Here,{right arrow over (ψ)}° is the full system mode counterpart of theaverage sector mode {right arrow over (ψ)}°⁽⁰⁾. Because mistuning issymmetric, the tuned and mistuned versions of the mode are nearlyidentical. Substituting {right arrow over (ψ)}° for {right arrow over(φ)}_(n) and pre-multiplying by {right arrow over (ψ)}°^(H) yields,(ω°_(ψ) ²+{right arrow over (ψ)}°^(H) ΔK{right arrow over (ψ)}°)−ω_(n)²(1+{right arrow over (ψ)}°^(H) ΔM{right arrow over (ψ)}°)=0  (10)These terms may be rearranged to isolate the frequency terms,{right arrow over (ψ)}°^(H)(ΔK−ω _(n) ² ΔM){right arrow over (ψ)}°=ω_(j)²−ω°_(ψ) ²  (11)Because the mistuning is symmetric, each sector contributes equally toequation (11). Thus, the contribution from the 0^(th) sector is,$\begin{matrix}{{{{\overset{->}{\psi}}^{{\bullet{(0)}}H}\left( {{\Delta\quad K} - {\omega_{n}^{2}\Delta\quad M}} \right)}{\overset{->}{\psi}}^{\bullet{(0)}}} = {\frac{1}{N}\left( {\omega_{j}^{2} - \omega_{\psi}^{\bullet 2}} \right)}} & (12)\end{matrix}$By factoring the frequency terms on the right-hand side of equation(12), it can be shown that $\begin{matrix}{{{{\overset{->}{\psi}}^{{\bullet{(0)}}H}\left( {{\Delta\quad K} - {\omega_{n}^{2}\Delta\quad M}} \right)}{\overset{->}{\psi}}^{\bullet{(0)}}} \approx \frac{2\omega_{\psi}^{\circ 2}{\Delta\omega}_{\psi}}{N}} & (13)\end{matrix}$where Δω_(ψ) is the fractional change in {right arrow over (ψ)}'snatural frequency due to mistuning, given byΔω_(ψ)=(ω_(ψ)−ω°_(ψ))/ω°_(ψ). Note that, by definition, as is a sectorfrequency deviation. Equation (13) can be substituted for the bracketedterms of equation (8), resulting in an expression that relates theelements of the sector “s” mistuning matrix to that sector's frequencydeviation, $\begin{matrix}{{\hat{A}}_{mn}^{(s)} = {\frac{2\omega_{m}^{*}\omega_{n}^{*}}{N}{\mathbb{e}}^{{\mathbb{i}}\quad{s{({n - m})}}\frac{2\pi}{N}}\Delta\quad\omega_{\psi}^{(s)}}} & (14)\end{matrix}$where the superscript on Δω_(ψ) Any is introduced to indicate that thefrequency deviation corresponds to the s^(th) sector. These sectorcontributions may be summed to obtain the elements of the mistuningmatrix, $\begin{matrix}{{\hat{A}}_{mn} = {2\omega_{m}^{0}{\omega_{n}^{0}\left\lbrack {\frac{1}{N}{\sum\limits_{s = 0}^{N - 1}{{\mathbb{e}}^{{\mathbb{i}}\quad{sp}\frac{2\pi}{N}}{\Delta\omega}_{\psi}^{(s)}}}} \right\rbrack}}} & (15)\end{matrix}$

1.3 The Simplified Form of the Fundamental Mistuning Model ModalEquation

The bracketed term in equation (15) is the discrete Fourier transform(DFT) of the sector frequency deviations. If one uses the dummy variablep to replace the quantity (n−m) in equation (15), then the p^(th) DFT ofthe sector frequency deviations is given by $\begin{matrix}{{\overset{\_}{\omega}}_{p} = \left\lbrack {\frac{1}{N}{\sum\limits_{s = 0}^{N - 1}{{\mathbb{e}}^{{\mathbb{i}}\quad{sp}\frac{2\pi}{N}}\Delta\quad\omega_{\psi}^{(s)}}}} \right\rbrack} & (16)\end{matrix}$where ω _(p) denotes the p^(th) DFT. By substituting equation (16) intoequation (15), Â may be expressed in the simplified matrix formÂ=2Ω° ΩΩ°  (17)where $\begin{matrix}{\overset{->}{\Omega} = \begin{bmatrix}{\overset{\_}{\omega}}_{0} & {\overset{\_}{\omega}}_{1} & \cdots & {\overset{\_}{\omega}}_{N - 1} \\{\overset{\_}{\omega}}_{N - 1} & {\overset{\_}{\omega}}_{0} & \cdots & {\overset{\_}{\omega}}_{N - 2} \\\vdots & \vdots & \quad & \vdots \\{\overset{\_}{\omega}}_{1} & {\overset{\_}{\omega}}_{2} & \cdots & {\overset{\_}{\omega}}_{0}\end{bmatrix}} & (18)\end{matrix}$Ω is a matrix which contains the discrete Fourier transforms of thesector frequency deviations. Note that Ω has a circulant form, and thuscontains only N distinct elements. Ω° is a diagonal matrix of the tunedsystem frequencies.

Substituting equation (17) into equation (3) produces the most basicform of the eigenvalue problem that may be solved to determine the modesand natural frequencies of the mistuned system.(Ω°²+2Ω° ΩΩ°){right arrow over (β)}_(j) ²{right arrow over(β)}_(j)  (19)

Equations (18) and (19) represent the functional form of the FundamentalMistuning Model. Here, Ω°² is a diagonal matrix of the nominal systemeigenvalues, ordered in accordance with the following equation.$\begin{matrix}\begin{Bmatrix}{\overset{->}{\phi}}_{0}^{\circ {(s)}} & {\overset{->}{\phi}}_{1}^{\circ {(s)}} & {\overset{->}{\phi}}_{2}^{\circ {(s)}} & \cdots & {\overset{->}{\phi}}_{\frac{N}{2}}^{\circ {(s)}} & {\overset{->}{\phi}}_{\frac{N}{2} + 1}^{\circ {(s)}} & \cdots & {\overset{->}{\phi}}_{N - 1}^{\circ {(s)}} \\(0) & \left( {1B} \right) & \left( {2B} \right) & \cdots & \left( \frac{N}{2} \right) & \left( {\left( {\frac{N}{2} - 1} \right)F} \right) & \cdots & \left( {1F} \right)\end{Bmatrix} & (20)\end{matrix}$where the second row in equation (20) indicates the nodal diameter anddirection of the corresponding mode that lies above it. “B” denotes abackward traveling wave, defined as a mode with a positive phase shiftfrom one sector to the next, and “F” denotes a forward traveling wave,defined as a mode with a negative phase shift from one sector to thenext. Note that the modes are numbered from 0 to N−1.

As mentioned before, the eigenvalues are equal to the squares of thenatural frequencies of the tuned system. This Ω°² matrix contains allthe nominal system information required to calculate the mistuned modes.Note that the geometry of the system does not directly enter into thisexpression. The term representing mistuning in equation (1), 2Ω°Ω Ω°, isa simple circulant matrix that contains the discrete Fourier transformsof the blade frequency deviations, pre- and post-multiplied by the tunedsystem frequencies.

The eigenvalues of equation (19) are the squares of the mistunedfrequencies, and the eigenvectors define the mistuned mode shapesthrough equation (2). Because the tuned modes have been limited to asingle family and appear in Φ° in a certain order, one can approximatelycalculate the distortion in the mistuned mode shapes without knowinganything specific about Φ°. The reason for this is the assumption thatall of the tuned system modes on the zero^(th) sector look nearly thesame, i.e. {right arrow over (φ)}°_(m) ⁽⁰⁾≈{right arrow over (φ)}°_(n)⁽⁰⁾. Further, when the tuned system modes are written in complex,traveling wave form, the amplitudes of every blade in a mode are thesame, but each blade has a different phase. Therefore, the part of themode corresponding to the s^(th) sector can be written in terms of thesame mode on the 0^(th) sector, multiplied by an appropriate phaseshift, i.e., $\begin{matrix}{{\overset{->}{\phi}}_{n}^{\circ {(s)}} = {{\mathbb{e}}^{{\mathbb{i}}\quad{sn}\frac{2\pi}{N}}{\overset{->}{\phi}}_{(n)}^{\circ {(0)}}}} & (21)\end{matrix}$where i=√{square root over (−1)}. Equation (21) implies that if thej^(th) mistune mode is given by {right arrow over (β)}_(j)=[β_(j0),β_(j1) . . . β_(j,N-1)]^(T) then the physical displacements of then^(th) blade in this mode are proportional to $\begin{matrix}{x_{n} = {\sum\limits_{m = 0}^{N - 1}{\beta_{jm}{\mathbb{e}}^{{\mathbb{i}}\quad{mn}\frac{2\pi}{N}}}}} & (22)\end{matrix}$

1.4 Numerical Results

A computer program was written to implement the theory presented insections (1.1) through (1.3). The program also incorporated a simplemodal summation algorithm to calculate the bladed disk's forcedresponse. The modal summation assumed constant modal damping. The basicmodal summation algorithm was chosen to benchmark the forced responsebecause a similar algorithm may be used as an option in the commercialfinite element analysis ANSYS® software, which was used as a benchmark.It is observed, however, that FMM may be used with more sophisticatedmethods for calculating the forced response, such as the state-spaceapproach used in a subset of nominal modes (SNM) analysis discussed inYang M.-T. and Griffin, J. H., 2001, “A Reduced Order Model of MistuningUsing a Subset of Nominal Modes,” Journal of Engineering for GasTurbines and Power, 123(4), pp. 893-900.

It is noted that when a beam-like blade model is used (to minimize thecomputational cost), FMM could accurately calculate a bladed disk'smistuned response based on only sector frequency deviations, withoutregard for the physical cause of the mistuning. However, in thediscussion below, a more realistic geometry is analyzed using FMM.

FIG. 6 illustrates an exemplary three dimensional (3D) finite elementmodel 22 of a high pressure turbine (HPT) blade-disk sector. There were24 sectors in the full system. This model was developed by approximatingthe features of an actual turbine blade and provided a reasonable testof FMM's ability to represent a realistic blade geometry. FIG. 7 showstuned system frequencies of the first families of modes of theblade-disk system modeled in FIG. 6. As can be seen from FIG. 7, thesystem of FIG. 6 had an isolated first bending family of modes withclosely spaced frequencies. As a benchmark, a finite element analysiswas performed of the full mistuned rotor using the ANSYS® software. Thebladed disk was mistuned by randomly varying the elastic moduli of theblades with a standard deviation that was equal to 1.5% of the tunedsystem's elastic modulus.

Then, an equivalent mass-spring model was constructed with onedegree-of-freedom per sector as described in Rivas-Guerra, A. J., andMignolet, M. P., 2001, “Local/Global Effects of Mistuning on the ForcedResponse of Bladed Disks,” ASME Paper 2001-GT-0289, International GasTurbine Institute Turbo Expo, New Orleans, La. Each mass was set tounity and the stiffness parameters were obtained through a least squaresfit of the tuned natural frequencies. FIG. 8 illustrates the tunedfrequencies of the fundamental family of modes of the system modeled inFIG. 6, along with the frequencies of the ANSYS® software and thebest-fit mass-spring model. It is noted that while the mass-spring modelwas able to capture the higher nodal diameter frequencies fairly well,it had great difficulty with the low nodal diameter frequencies. Thisdiscrepancy arises because the natural frequencies of the singledegree-of-freedom mass-spring system have the formω_(n) =√{square root over ([k+4k_(c) sin² (nπ/N)]/m)}  (23)where m is the blade mass, k and k_(c) are the base stiffness andcoupling stiffness, n is the nodal diameter of the mode, and N is thenumber of blades. However, the actual frequencies of the finite elementmodel have a significantly different shape when plotted as a function ofnodal diameters. In contrast, FMM takes the actual finite elementfrequencies as input parameters, and therefore it matches the tunedsystem's frequencies exactly.

The mass-spring model was then mistuned by adjusting the base stiffnessof the blades to correspond to the modulus changes used in the finiteelement model. The mistuned modes and forced response were thencalculated with both FMM and the mass-spring model, and compared withthe finite element results using the ANSYS® software. FIGS. 9 (a) and(b) depict representative results of using FMM with a realistic mistunedbladed disk modeled in FIG. 6. As can be seen from FIG. 9(a), themistuned frequencies predicted by FMM and ANSYS® software were quitesimilar. However the mass-spring model had some significantdiscrepancies, especially in the low frequency modes. FMM and ANSYS®software also predicted essentially the same mistuned mode shapes as canbe seen from FIG. 9(b). In contrast, the mass-spring model performedpoorly when matching the finite element mode shapes, even on modes whosefrequencies were accurately predicted. For example, the mode plotted inFIG. 9(b) corresponded to the 18^(th) frequency. From FIG. 9(a), it isseen that the mass-spring model accurately predicted this frequency.However, it is clear from FIG. 9(b) that the mass-spring model still dida poor job of matching the finite element mode shapes.

The predicted modes were then summed to obtain the system's forcedresponse to various engine order excitations. As noted before, gasturbine engines are composed of a series of bladed disks (see, forexample, FIG. 1). When a bladed disk is operating in an engine, it issubjected to pressure loading from the flow field which excites theblades. As the flow progresses through the engine, it passes oversupport struts, inlet guide vanes, and other stationary structures whichcause the pressure to vary circumferentially. Therefore, the excitationforces are periodic in space when considered from a stationary referenceframe. As a periodic excitation, the pressure variations can bespatially decomposed into a Fourier series. Each harmonic componentdrives the system with a traveling wave at a frequency given by theproduct of its harmonic number and the rotation speed. The harmonicnumber of the excitation is typically referred to as the Engine Order,and corresponds physically to the number of excitation periods perrevolution. Each of the engine order excitations may be generallyconsidered separately, because they drive the system at differentfrequencies.

FIGS. 10(a) and (b) show a representative case of the blade amplitudesas a function of excitation frequency for a 7^(th) engine orderexcitation predicted by the mass-spring model, ANSYS® software, and FMM.For clarity, only the high responding, median responding, and lowresponding blades are plotted. It is again seen that the mass-springmodel provided a poor prediction of the system's forced response.However, the results from FMM agreed well with those computed by ANSYS®software, as shown in FIG. 10(b). The prediction by FMM of the highestblade response differed from that predicted by ANSYS® software by only1.6%. Thus, FMM may be used to provide accurate predictions of the modeshapes and forced response of a turbine blade with a realistic geometry.

1.5 Other Considerations

From the foregoing discussion, it is seen that the Fundamental MistuningModel was derived from the Subset of Nominal Modes theory by applyingthree simplifying assumptions: only a single, isolated family of modesis excited; the strain energy of that family's modes is primarily in theblades; and the family's frequencies are closely spaced. In addition,one corollary of these assumptions is that the blade's motion looks verysimilar among all modes in the family. As demonstrated in the previoussection, FMM works quite well when these assumptions are met. However,these ideal conditions are usually found only in the fundamental modesof a rotor. The higher frequency families are often clustered closetogether, have a significant amount of strain energy in the disk, andspan a large frequency range. Furthermore, veerings are quite common,causing a family's modes to change significantly from one nodal diameterto the next. Therefore, there may be situations where FMM may not workeffectively in high frequency regions.

The realistic HPT model of FIG. 6 may be used to further study FMM'sperformance, without the need carefully assign modes to families.Therefore, some crossings shown in FIG. 7 may in fact be veerings.However, because such errors are easily made in practice, it is usefulto include them in the study. For reference, four mode families arenumbered along the right side of FIG. 7. It is noted that except for thefundamental modes, the families (in FIG. 7) undergo a significantfrequency increase between 0 and 6 nodal diameters. The steep slopes inthis region suggest that the modes have a large amount of strain energyin the disk. Furthermore, the high modal density in this area makes itlikely that some modes were assigned to the wrong family. Therefore, themodes of a single family may likely change significantly from one nodaldiameter to the next. To show this behavior, FIG. 11 illustrates theleading edge blade tip displacements for the third family of modes shownin FIG. 7. FIG. 11 shows how the circumferential (O) and out-of-plane(z) motion of the blade tip's leading edge changes from the 0 nodaldiameter mode to the 12 nodal diameter mode. Observe that the 0 and zcomponents of the mode shape change significantly between 0 and 6 nodaldiameters. In such case, the assumptions of FMM are violated throughoutthese low nodal diameter regions. Thus, FMM may not accurately predictthe mistuned frequencies or shapes of these modes. As a result, FMM maynot provide accurate forced response predictions when these modes areheavily excited.

To illustrate FMM performance in such situations, FMM was used topredict the forced response of families 2, 3, and 4 to a 2^(nd)engine-order excitation because that engine order would primarily excitethe low nodal diameter modes of each family, and those modes violate theassumptions of FMM. The FMM predictions were compared with finiteelement results calculated in ANSYS® software. FIGS. 12(a)-(c)illustrate FMM and ANSYS® software predictions of blade amplitude as afunction of excitation frequency for a 2^(nd) engine order excitation of2^(nd), 3^(rd), and 4^(th) families respectively. For clarity, each plotin FIGS. 12(a)-(c) shows only the low responding blade, the medianresponding blade, and the high responding blade. As expected, the FMMresults differed significantly from the ANSYS® software response in bothpeak amplitudes and overall shape of the response. Thus, when a modelies in a region where there is uncertainty as to what family a modebelongs, veering, or high slopes on the frequency vs. nodal diameterplot, FMM may not always accurately predict its mistuned frequency ormode shape. That is, FMM may not work effectively for engine orders thatexcite modes in these regions.

However, there are regions in high frequency modes where FMM may performquite well. It is seen from the Frequency vs. Nodal Diameter plot inFIG. 7 that the slopes over the high nodal diameter regions are verysmall, indicating that the modes have most of their strain energy in theblades. Furthermore, the flat regions are well isolated from otherfamilies of modes. Therefore, the blade's motion is very similar fromone nodal diameter to the next. This can be seen in FIG. 11, whichindicates that the 0 and z components of the blade tip motion remainfairly constant over the higher nodal diameter regions. In that case,the FMM assumptions are satisfied for these high nodal diameter modes,and FMM may capture the physical behavior of these modes better than itdid for low engine orders.

To illustrate FMM performance in the situation described in thepreceding paragraph, FMM was used to predict the forced response offamilies 2, 3, and 4 to a 7^(th) engine order excitation. The FMMresults were compared against a finite element benchmark performed inANSYS® software. FIGS. 13(a)-(c) illustrate FMM and ANSYS® softwarepredictions of blade amplitude as a function of excitation frequency fora 7^(th) engine order excitation of 2^(nd), 3^(rd), and 4^(th)familiesrespectively. For clarity, each plot in FIGS. 13(a)-(c) shows only thelow responding blade, the median responding blade, and the highresponding blade. In all three cases in FIG. 13, the FMM predictionscaptured the overall shape of the response curves as well as the peakamplitudes to within 6% of the ANSYS® software performance. Therefore,for this test case, it is observed that when a mode lies in a flatregion at the upper end of a Frequency vs. Nodal Diameter plot, itsresponse can be reasonably well predicted by FMM.

[2] SYSTEM IDENTIFICATION METHODS

It is seen from the discussion hereinbefore that the FundamentalMistuning Model provides a simple, but accurate method for assessing theeffect of mistuning on forced response, generally in case of an isolatedfamily of modes. However, FMM can be used to derive more complex reducedorder models to analyze mistuned response in regions of frequencyveering, high modal density and cases of disk dominated modes. Thesecomplex models may not necessarily be limited to an isolated family ofmodes.

The following description of system identification is based theFundamental Mistuning Model. As a result, the FMM based identificationmethods (FMM ID) (discussed below) may be easy to use and may requirevery little analytical information about the system, e.g., no finiteelement mass or stiffness matrices may be necessary. There are two formsof FMM ID methods discussed below: a basic version of FMM ID thatrequires some information about the system properties, and a somewhatmore advanced version that is completely experimentally based. The basicFMM ID requires the nominal frequencies of the tuned system as input.The nominal frequencies of the tuned system (i.e., natural frequenciesof a tuned system with each sector being identical) may be calculatedusing a finite element analysis of a single blade-disk sector withcyclic symmetric boundary conditions applied to the disk. Then, given(experimental) measurements of a limited number of mistuned modes andfrequencies, basic FMM ID equations solve for the mistuned frequency ofeach sector. It is noted that the modes required in basic FMM ID are thecircumferential modes that correspond to the tip displacement of eachblade around the wheel or disk.

The advanced form of FMM ID uses (experimental) measurements of somemistuned modes and frequencies to determine all of the parameters inFMM, i.e. the frequencies that the system would have if it were tuned aswell as the mistuned frequency of each sector. Thus, the tuned systemfrequencies determined from the second method (i.e., advanced FMM ID)can also be used to validate finite element models of the nominalsystem.

2.1 Basic FMM ID Method

As noted before, the basic method uses tuned system frequencies alongwith measurements of the mistuned rotor's system modes and frequenciesto infer mistuning.

2.1.1 Inversion of FMM Equation

The FMM eigenvalue problem is given by equation (19), which isreproduced below.(Ω°²+2Ω° ΩΩ°) β _(j)=ω_(j) ² β _(j)  (24)The eigenvector of this equation, {right arrow over (β)}_(j), containsweighting factors that describe the j^(th) mistuned mode as a sum oftuned modes. The corresponding eigenvalue, ω_(j) ², is the j^(th) mode'snatural frequency squared. The matrix of the eigenvalue problem containstwo terms, Ω° and Ω. Ω° is a diagonal matrix of the tuned systemfrequencies, ordered by ascending inter-blade phase angle of theircorresponding mode. The notation a Ω°² is shorthand for Ω°^(T)Ω°, whichresults in a diagonal matrix of the tuned system frequencies squared.The matrix Ω contains the discrete Fourier transforms (DFT) of thesector frequency deviations.

As discussed earlier, FMM treats the rotor's mistuning as a knownquantity that it uses to determine the system's mistuned modes andfrequencies. However, if the mistuned modes and frequencies are treatedas known parameters, the inverse problem could be solved to determinethe rotor's mistuning. This is the basis of FMM ID methods.

The following describes manipulation of the FMM equation of motion tosolve for the mistuning in the rotor. Thus, in equation (24), allquantities are treated as known except Ω, which describes the system'smistuning. Subtracting the Ω°² term from both sides of equation (24) andregrouping terms yields2Ω° Ω[Ω°{right arrow over (β)}_(j)]=(ω_(j) ² I−Ω° ²){right arrow over(β)}_(j)  (25)The bracketed quantity on the left-hand side of equation (25) contains aknown vector, which may be denoted as {right arrow over (γ)}_(j),{right arrow over (γ)}_(j)=Ω°{right arrow over (β)}_(j)  (26)Thus, {right arrow over (γ)}_(j) contains the modal weighting factors,{right arrow over (β)}_(j) scaled on an element-by-element basis bytheir corresponding natural frequencies. Substituting {right arrow over(γ)}_(j) into equation (25) yields2Ω°[ Ω{right arrow over (γ)}_(j)]=(ω_(j) ² I−Ω° ²){right arrow over(β)}_(j)  (27)After some algebra, it can be shown that the product in the bracket inequation (27) may be rewritten in the formΩ{right arrow over (γ)}_(j)=Γ_(j) {right arrow over (ω)}  (28)where the vector {right arrow over (ω)} equals [ ω ₀, ω ₁ . . . ω_(N-1)]^(T). The matrix Γ_(j) is composed from the elements in {rightarrow over (γ)}_(j) and has the form $\begin{matrix}{\Gamma_{j} = \begin{bmatrix}{\overset{\_}{\gamma}}_{j\quad 0} & {\overset{\_}{\gamma}}_{j\quad 1} & \cdots & {\overset{\_}{\gamma}}_{j{({N - 1})}} \\{\overset{\_}{\gamma}}_{j\quad 1} & {\overset{\_}{\gamma}}_{j\quad 2} & \cdots & {\overset{\_}{\gamma}}_{j\quad 0} \\\vdots & \vdots & \quad & \vdots \\{\overset{\_}{\gamma}}_{j{({N - 1})}} & {\overset{\_}{\gamma}}_{j\quad 0} & \cdots & {\overset{\_}{\gamma}}_{j{({N - 2})}}\end{bmatrix}} & (29)\end{matrix}$where γ_(jn) denotes the n^(th) element of the vector {right arrow over(γ)}_(j); the {right arrow over (γ)}_(j) elements are numbered from 0 toN−1. Note that each column of Γ_(j) is the negative permutation of theprevious column.

Substituting equation (28) into (27) produces an expression in which thematrix of mistuning parameters, Ω, has been replaced by a vector ofmistuning parameters, {right arrow over (ω)}.2Ω°Γ_(j) {right arrow over (ω)}=(ω_(j) ² I−Ω° ²){right arrow over(β)}_(j)  (30)Pre-multiplying equation (30) by (2Ω°Γ_(j))⁻¹ would solve thisexpression for the DFT (Discrete Fourier Transform) of the rotor'smistuning. Furthermore, the vector {right arrow over (ω)} can then berelated to the physical sector mistuning through an inverse discreteFourier transform. However, equation (30) only contains data from onemeasured mode and frequency. Therefore, error in the mode's measurementmay result in significant error in the predicted mistuning. To minimizethe effects of measurement error, multiple mode measurements may beincorporated into the solution for the mistuning. Equation (30) may beconstructed for each of the M measured modes, and the modes may becombined into the single matrix expression, $\begin{matrix}{{\begin{bmatrix}{2\Omega^{\circ}\Gamma_{1}} \\{2\Omega^{\circ}\Gamma_{2}} \\\vdots \\{2\Omega^{\circ}\Gamma_{m}}\end{bmatrix}\overset{\overset{->}{\_}}{\omega}} = \begin{bmatrix}{\left( {{\omega_{1}^{2}I} - \Omega^{\circ 2}} \right){\overset{->}{\beta}}_{1}} \\{\left( {{\omega_{2}^{2}I} - \Omega^{\circ 2}} \right){\overset{->}{\beta}}_{2}} \\\vdots \\{\left( {{\omega_{m}^{2}I} - \Omega^{\circ 2}} \right){\overset{->}{\beta}}_{m}}\end{bmatrix}} & (31)\end{matrix}$For brevity, equation (31) may be rewritten as{tilde over (L)} {right arrow over (ω)}= {right arrow over (r)}  (32)where {tilde over (L)} is the matrix on the left-hand side of theexpression, and {right arrow over (r)} is the vector on the right-handside. The “˜” is used to indicate that these quantities are composed byvertically stacking a set of sub-matrices or vectors.

It is noted that the expression in equation (32) is an overdeterminedset of equations. Therefore, it may not be possible to solve for {rightarrow over (ω)} by direct inverse. However, one can obtain a leastsquares fit to the mistuning, i.e. $\begin{matrix}{\overset{\overset{->}{\_}}{\omega} = {{Lsp}\left\{ {\overset{\sim}{L},\overset{\overset{\sim}{->}}{r}} \right\}}} & (33)\end{matrix}$Equation (33) produces the vector {right arrow over (ω)} which best-fitsall the measured data. Therefore, the error in each measurement iscompensated for by the balance of the data. The vector {right arrow over(ω)} can then be related to the physical sector mistuning through theinverse transform, $\begin{matrix}{{\Delta\omega}_{\psi}^{(s)} = {\sum\limits_{p = 0}^{N - 1}{{\mathbb{e}}^{{- {\mathbb{i}}}\quad{sp}\frac{2\pi}{N}}{\overset{\_}{\omega}}_{p}}}} & (34)\end{matrix}$where Δω_(ψ) ^((s)) is the sector frequency deviation of the s^(th)sector. The following section describes how equations (33) and (34) canbe applied to determine a rotor's mistuning.

2.1.2 Experimental Application of Basic FMM ID

To solve equation (33) and (34) for the sector mistuning, one must firstconstruct {tilde over (L)} and {right arrow over (r)} from the tunedsystem frequencies and the mistuned modes and frequencies. The tunedsystem frequencies can be calculated through finite element analysis ofa tuned, cyclic symmetric, single blade/disk sector model. However, themistuned modes and frequencies must be obtained experimentally.

The modes used by basic FMM ID are circumferential modes, correspondingto the tip displacement of each blade on the rotor. In case of isolatedfamilies of modes, it may be sufficient to measure the displacement ofonly one point per blade. In practice, modes and frequencies may beobtained by first measuring a complete set of frequency responsefunctions (FRFs). Then, the modes and frequencies may be extracted fromthe FRFs using modal curve fitting software.

The mistuned frequencies obtained from the measurements appearexplicitly in the basic FMM ID equations as ω_(j). However, the mistunedmodes enter into the equations indirectly through the modal weightingfactors {right arrow over (β)}_(j). Each vector {right arrow over(β)}_(j is) obtained by taking the inverse discrete Fourier transform ofthe corresponding single point-per-blade mode, i.e., $\begin{matrix}{\beta_{jn} = {\sum\limits_{m = 0}^{N - 1}{\phi_{jm}{\mathbb{e}}^{{- {\mathbb{i}}}\quad{mn}\frac{2\pi}{N}}}}} & (35)\end{matrix}$The quantities obtained from equation (35) may then be used with thetuned system frequencies to construct {tilde over (L)} and {right arrowover (r)} as outlined hereinbefore. Finally, equations (33) and (34) maybe solved for the sector mistuning. This process is demonstrated throughthe two examples in the following section.

2.1.3 Numerical Examples for Basic FMM ID

The first example considers an integrally bladed compressor whose bladesare geometrically mistuned. The sector frequency deviations identifiedby basic FMM ID are verified by comparing them with values directlydetermined by finite element analyses (FEA). The second examplehighlights basic FMM ID's ability to detect mistuning caused byvariations at the blade-disk interface. This example considers acompressor in which all the blades are identical, however they aremounted on the disk at slightly different stagger angles. The mistuningcaused by the stagger angle variations is determined by FMM ID andcompared with the input values.

2.1.3.1 Geometric Blade Mistuning

FIG. 14 represents an exemplary finite element model 26 of a twentyblade compressor. Although the airfoils on this model are simply flatplates, the rotor design reflects the key dynamic behaviors of a modern,integrally bladed compressor. The rotor was mistuned through acombination of geometric and material property changes. Approximatelyone-third of the blades were mistuned through length variations,one-third through thickness variations, and one-third through elasticmodulus variations. The magnitudes of the variations were chosen so thateach form of mistuning would contribute equally to a 1.5% standarddeviation in the sector frequencies.

A finite element analysis of the tuned rotor was first performed togenerate its nodal diameter map. FIG. 15 illustrates the naturalfrequencies of the tuned compressor modeled in FIG. 14, i.e., the tunedrotor's nodal diameter map. It is observed from FIG. 15 that the lowestfrequency family of first bending modes was isolated (as denoted by therectangle portion 27) for the basic FMM ID analysis. The sectormistuning of this rotor was then determined through two differentmethods: finite element analyses (FEA) of the mistuned sectors using thecommercially available ANSYS finite element code, and basic FMM ID.

The finite element calculations serve as a benchmark to assess theaccuracy of the basic FMM ID method. In the benchmark, a finite elementmodel was made for each mistuned blade. In the model the blade wasattached to a single disk sector. The frequency change in the mistunedblade/disk sector was then calculated with various cyclic symmetricboundary conditions applied to the disk. It was found that the phaseangle of the cyclic symmetric constraint had little effect on thefrequency change caused by blade mistuning. The values described hereinwere for a disk phase constraint of 90 degrees, i.e., for the five nodaldiameter mode.

A finite element model of the full, mistuned bladed disk was alsoconstructed and used to compute its mistuned modes and naturalfrequencies. The modes and frequencies were used as input data for basicFMM ID. In another embodiment, the mistuned modes and frequencies may beobtained through a modal fit of the rotor's frequency responsefunctions. Typically, the measurements may not detect modes that have anode point at the excitation source. To reflect this phenomenon, allmistuned modes that had a small response at blade one were eliminated.This left 16 modes and natural frequencies to apply to basic FMM ID.

The mistuned modes and frequencies were combined with the tuned systemfrequencies of the fundamental mode family to construct the basic FMM IDequations (31). These equations were solved using a least squares fit.The solution was then converted to the physical sector frequencydeviations through the inverse transform given in equation (34).

FIG. 16 shows the comparison between the sector mistuning calculateddirectly by finite element simulations of each mistuned blade/sector andthe mistuning identified by basic FMM ID. As is seen from FIG. 16, thetwo results are in good agreement.

2.1.3.2 Stagger Angle Mistuning

One of the differences between basic FMM ID and other mistuningidentification methods is its measure of mistuning. Basic FMM ID uses afrequency quantity that characterized the mistuning of an entireblade-disk sector, whereas other methods in the literature considermistuning to be confined to the blades as can be seen, for example, fromthe discussion in Judge, J. A., Pierre, C., and Ceccio, S. L., 2002,“Mistuning Identification in Bladed Disks,” Proceedings of theInternational Conference on Structural Dynamics Modeling, MadeiraIsland, Portugal. The sector frequency approach used by FMM not onlyidentifies the mistuning in the blades, but it also captures themistuning in the disk and the blade-disk interface. To highlight thiscapability, the following example considers a rotor in which the bladesare identical except they are mounted on the disk with slightlydifferent stagger angles. FIG. 17 schematically illustrates a rotor 29with exaggerated stagger angle variations as viewed from above.

In case of the compressor 26 in FIG. 14, its rotor was mistuned byrandomly altering the stagger angle of each blade with a maximumvariation of ±4°. Otherwise the blades were identical. The modes of thesystem were then calculated using the ANSYS® finite element code. FIG.18 shows a representative mistuned mode caused by stagger anglemistuning of the rotor in FIG. 14. It is seen in FIG. 18 that the modewas localized (in the higher blade number region), indicating thatvarying the stagger angles does indeed mistune the system.

The mistuned modes and frequencies calculated by ANSYS® software werethen used to perform a basic FMM ID analysis of the mistuning. Theresulting sector frequency deviations are plotted as the solid line inFIG. 19, which illustrates a comparison of mistuning determination frombasic FMM ID and the variations in the stagger angles. The circles inFIG. 19 correspond to the stagger angle variations applied to eachblade. The vertical axes in FIG. 19 have been scaled so that the maximumfrequency and angle variation data points (blade 14) are coincident.This was done to highlight the fact that the stagger angle variationsare proportional to the sector frequency deviations detected by basicFMM ID. Thus, not only can basic FMM ID substantially accurately detectmistuning in the blades, as illustrated in the previous example, but itcan also substantially accurately detect other forms of mistuning suchas variation in the blade stagger angle.

2.2 Advanced FMM ID Method

As discussed before, the basic FMM ID method provides an effective meansof determining the mistuning in an IBR. The basic FMM ID techniquerequires a set of simple vibration measurements and the naturalfrequencies of the tuned system. However, at times neither the tunedsystem frequencies nor a finite element model from which to obtain themare available to determine an IBR's mistuning. Furthermore, even if afinite element model is available, there is often concern as to howaccurately the model represents the actual rotor. Therefore, thefollowing describes an alternative FMM ID method (advanced FMM ID) thatdoes not require any analytical data. Advanced FMM ID requires only alimited number of mistuned modes and frequency measurements to determinea bladed disk's mistuning. Furthermore, the advanced FMM ID method alsoidentifies the bladed disk's tuned system frequencies. Thus, advancedFMM ID not only serves as a method of identifying mistuning of thesystem, but can also provide a method of corroborating the finiteelement model of the tuned system

2.2.1 Advanced FMM ID Theory

Advanced FMM ID may be derived from the basic FMM ID equations. Recallthat a step in the development of the basic FMM ID theory was totransform the mistuning matrix Ω into a vector form. Once the mistuningwas expressed as a vector, it could then be calculated using standardmethods from linear algebra. A similar approach is used below to solvefor the tuned system frequencies. However, the resulting equations arenonlinear, and require a more sophisticated solution approach.

2.2.1.1 Development of Nonlinear Equations

Consider the basic FMM ID equation (30). Moving the Ω°² term to theleft-hand side of the equation, the expression becomesΩ°²{right arrow over (β)}_(j)2Ω°Γ_(j) {right arrow over (ω)}=ω_(j)²{right arrow over (β)}_(j)  (36)It is assumed that from measurement of the mistuned modes andfrequencies, {right arrow over (β)}_(j) and ω_(j) in equation (36) areknown. All other quantities are unknown. It is noted that although Γ_(j)is not known, the matrix contains elements from {right arrow over(β)}_(j). Therefore, some knowledge of the matrix is available.

After some algebra, one can show that the term Ω°²{right arrow over(β)}_(j) in equation (36) may be re-expressed asΩ°²{right arrow over (β)}_(j)=B_(j){right arrow over (λ)}°  (37)where {right arrow over (λ)}° is a vector of the tuned frequenciessquared, and B_(j) is a matrix composed from the elements of {rightarrow over (μ)}_(j). If η is defined to be the maximum number of nodaldiameters on the rotor, i.e. η=N/2 if N is even or (N−1)/2 if N is odd,then {right arrow over (λ)}° is given by $\begin{matrix}{{\overset{->}{\lambda}{^\circ}} = \begin{bmatrix}\omega_{0\quad{ND}}^{\circ 2} \\\omega_{1\quad{ND}}^{\circ 2} \\\vdots \\\omega_{\eta\quad{ND}}^{\circ 2}\end{bmatrix}} & (38)\end{matrix}$For N even, the matrix B_(j) has the form $\begin{matrix}{B_{j} = \begin{bmatrix}\beta_{j\quad 0} & \quad & \quad & \quad & \quad \\\quad & \beta_{j\quad 1} & \quad & \quad & \quad \\\quad & \quad & \beta_{j\quad 2} & \quad & \quad \\\quad & \quad & \quad & ⋰ & \quad \\\quad & \quad & \quad & \quad & \beta_{jm} \\\quad & \quad & \quad & \ddots & \quad \\\quad & \quad & {\quad\beta_{j\quad 2}} & \quad & \quad \\\quad & \beta_{j\quad 1} & \quad & \quad & \quad\end{bmatrix}} & (39)\end{matrix}$A similar expression can be derived for N odd.

Substituting equation (37) into (36) and regrouping the left-hand sideresults in a matrix equation for the tuned frequencies squared and thesector mistuning, $\begin{matrix}{{\begin{bmatrix}B_{j} & {2\Omega^{\circ}\Gamma_{j}}\end{bmatrix}\begin{bmatrix}\overset{->}{\lambda} \\\overset{\overset{->}{\_}}{\omega}\end{bmatrix}} = {\omega_{j}^{2}{\overset{->}{\beta}}_{j}}} & (40)\end{matrix}$Equation (40) contains information from only one of the M measured modesand frequencies. However, equation (40) can be constructed for eachmeasured mode, and combined into the single matrix expression$\begin{matrix}{{\begin{bmatrix}B_{1} & {2\Omega^{\circ}\Gamma_{1}} \\B_{2} & {2\Omega^{\circ}\Gamma_{2}} \\\vdots & \vdots \\B_{M} & {2\Omega^{\circ}\Gamma_{M}}\end{bmatrix}\begin{bmatrix}\overset{->}{\lambda} \\\overset{\overset{->}{\_}}{\omega}\end{bmatrix}} = \begin{bmatrix}{\omega_{1}^{2}{\overset{->}{\beta}}_{1}} \\{\omega_{2}^{2}{\overset{->}{\beta}}_{2}} \\\vdots \\{\omega_{M}^{2}{\overset{->}{\beta}}_{M}}\end{bmatrix}} & (41)\end{matrix}$Thus, equation (41) represents a single expression that incorporates allof the measured data. For brevity, equation (41) is rewritten as$\begin{matrix}{{\begin{bmatrix}\overset{\sim}{B} & {2\left( \overset{\sim}{\Omega^{\circ}\Gamma} \right)}\end{bmatrix}\begin{bmatrix}\overset{->}{\lambda} \\\overset{\overset{->}{\_}}{\omega}\end{bmatrix}} = \overset{\overset{\sim}{->\prime}}{r}} & (42)\end{matrix}$where {tilde over (B)} is the stacked matrix of B_(j), the term (

) is the stacked matrix of Ω°Γ_(j), and {right arrow over (r)} is theright-hand side of equation (41).

An additional constraint equation may be required because the equations(42) are underdetermined. To understand the cause of this indeterminacy,consider a rotor in which each sector is mistuned the same amount. Dueto the symmetry of the mistuning, the rotor's mode shapes will stilllook tuned, but its frequencies will be shifted. If one has no priorknowledge of the tuned system frequencies, there may not be any way todetermine that the rotor has in fact been mistuned. The same difficultyarises in solving equation (42) because there may not be any way todistinguish between a mean shift in the mistuning and a correspondingshift in the tuned system frequencies. To eliminate this ambiguity,mistuning may be defined so that it has a mean value of zero.

Mathematically, a zero mean in the mistuning translates to prescribingthe first element of {right arrow over (ω)} to be zero. With theaddition of this constraint, equation (42) takes the form$\begin{matrix}{{\begin{bmatrix}\overset{\sim}{B} & {2\left( \overset{\sim}{\Omega^{\circ}\Gamma} \right)} \\0 & \overset{->}{c}\end{bmatrix}\begin{bmatrix}{\overset{->}{\lambda}}^{*} \\\overset{\overset{->}{\_}}{\omega}\end{bmatrix}} = \begin{bmatrix}\overset{\overset{\sim}{->}}{r} \\0\end{bmatrix}} & (43)\end{matrix}$where {right arrow over (c)} is a row vector whose first element is 1and whose remaining elements are zero.

2.2.1.2 Iterative Solution Method

If the term (

) in equation (43) were known, then a least squares solution could beobtained for the tuned eigenvalues {right arrow over (λ)}° and the DFTof the sector mistuning {right arrow over (ω)}. However, because (

) is based in part on the unknown quantities {right arrow over (λ)}°,the equations in (43) are nonlinear. Therefore, an alternative solutionmethod may be devised. In a solution described below, an iterativeapproach is used to solve the equations in (43).

In iterative form, the least squares solution to equation (43) can bewritten as $\begin{matrix}{\begin{bmatrix}{\overset{->}{\lambda}}^{*} \\\overset{\overset{->}{\_}}{\omega}\end{bmatrix}_{(k)} = {{Lsq}\left\{ {\begin{bmatrix}\overset{\sim}{B} & {2\left( \overset{\sim}{\Omega^{\circ}\Gamma} \right)_{({k - 1})}} \\0 & \overset{->}{c}\end{bmatrix},\begin{bmatrix}\overset{\overset{\sim}{->}}{r} \\0\end{bmatrix}} \right\}}} & (44)\end{matrix}$where the subscripts indicate the iteration number. For each iteration,a new matrix (

) may be constructed based on the previous iteration's solution for{right arrow over (λ)}°. This process may be repeated until a convergedsolution is obtained. With a good initial guess, this method maytypically converge within a few iterations.

To identify a good initial guess, in case of analyzing an isolatedfamily of modes, it is observed that generally the frequencies ofisolated mode families tend to span a fairly small range. Therefore, onegood initial guess is to take all of the tuned frequencies to be equalto one another, and assigned the value of the mean tuned frequency, i.e.{right arrow over (λ)}°₍₀₎=ω°_(avg) ²  (45)However, the value of ω°_(avg) is not known and therefore cannot bedirectly applied to equation (44). Consequently, equation (43) may beslightly modified to incorporate the initial guess defined by equation(45). In equation (43), if the tuned frequencies are taken to be equalto ω°_(avg), then the term (

) may be expressed as $\begin{matrix}{\left( \overset{\sim}{\Omega^{\circ}\Gamma} \right) = {\omega_{avg}^{\circ}\overset{\sim}{\Gamma}}} & (46)\end{matrix}$where {tilde over (Γ)} is the matrix formed by vertically stacking the MΓ_(j) matrices.

The matrix Γ_(j) is also related to the tuned frequencies. As a result,the elements of each matrix Γ_(j) simplify to the form ω°_(avg)β_(jn).This allows one to rewrite Γ_(j) asΓ_(j)=ω°_(avg)Z_(j)  (47)where Z_(j) is composed of the elements β_(jn) arranged in the samepattern as the γ_(jn) elements shown in equation (29). Thus,consolidating all ω°_(avg) terms, equation (46) can be written as$\begin{matrix}{\left( \overset{\sim}{\Omega^{\circ}\Gamma} \right) = {\omega_{avg}^{\circ 2}\overset{\sim}{Z}}} & (48)\end{matrix}$where {tilde over (Z)} is the stacked form of the Z_(j) matrices.

Substituting equation (48) into equation (43) and regrouping termsresults in the expression $\begin{matrix}{{\begin{bmatrix}\overset{\sim}{B} & {2\overset{\sim}{Z}} \\0 & \overset{->}{c}\end{bmatrix}\begin{bmatrix}{\overset{->}{\lambda}}^{*} \\{\omega_{avg}^{\circ 2}\overset{\overset{->}{\_}}{\omega}}\end{bmatrix}} = \begin{bmatrix}\overset{\overset{\sim}{->}}{r} \\0\end{bmatrix}} & (49)\end{matrix}$Note that the ω°_(avg) ² term was grouped with the vector {right arrowover (ω)}. Thus, all the unknown expressions are consolidated into thesingle vector on the left-hand side of equation (49). These quantitiescan be solved through a least squared fit of the equations. Thisrepresents the 0^(th) iteration of the solution process. The {rightarrow over (λ)}° terms of the solution may then be used as an initialguess for the first iteration of equation (44).

In practice, the mistuned modes and frequencies may be measured usingthe technique described for basic FMM ID in Section 2.1.3.2. The nextsection presents a numerical example that demonstrates the ability ofthe advanced FMM ID method to identify the frequencies of the tunedsystem as well as mistuned sector frequencies.

2.2.2 Numerical Test Case for Advanced FMM ID

This section presents a numerical example of the advanced FMM ID methodthat identifies the tuned system frequencies as well as the mistuning.This example uses the same geometrically mistuned compressor model 26(FIG. 14) as that used for the basic FMM ID method. The tuned systemfrequencies and sector mistuning identified by advanced FMM ID are thencompared with finite element results.

The modes and natural frequencies of the mistuned bladed disk werecalculated using a finite element model of the mistuned system. Thephysical modes were then converted to vectors of modal weightingfactors, {right arrow over (β)}, through equation (35). The weightingfactors were used to form the elements of equation (49) which was solvedto obtain an initial estimate of the tuned system frequencies. Thisinitial estimate was used as an initial guess to iteratively solveequation (44). The solution vector contained two parts: a vector of thetuned system frequencies squared, and a vector of the DFT of the sectorfrequency deviations. The sector mistuning was converted to the physicaldomain using the inverse transform in equation (34).

The resulting sector frequency deviations were compared with thebenchmark finite element analysis (FEA) values. FIG. 20 depicts acomparison of mistuning predicted using advanced FMM ID with thatobtained using the finite element analysis (FEA). The results in FIG. 20were obtained using the same procedure as that discussed in section2.1.3 above. FIG. 21 shows a comparison of the tuned frequenciesidentified by advanced FMM ID and those computed directly with thefinite element model (i.e., FEA). In each of FIGS. 20 and 21, theresults obtained using advanced FMM ID were in good agreement with thosefrom FEA.

[3] SYSTEM IDENTIFICATION: APPLICATION

FIG. 22 illustrates an exemplary setup 32 to measure transfer functionsof test rotors and also to verify various FMM ID methods discussedhereinbefore. As discussed earlier, the advanced FMM ID method uses themeasurements of the mistuned rotor's system modes and naturalfrequencies. The term “system mode” is used herein to refer to the tipdisplacement of each blade as a function of blade's angular position.The system modes may be obtained using a standard modal analysisapproach: measure the bladed disk's transfer functions, and thencurve-fit the transfer functions to obtain the mistuned modes andnatural frequencies. The setup 32 in FIG. 22 may be used to performstandard transfer function measurements. As illustrated in FIG. 22, therotor to be tested (rotor 34) may be placed on a foam pad 36 toapproximate a free boundary condition. Then, one of the rotor blades maybe excited over the frequency range of interest using an excitationsource 38 (for example, a function generator coupled to an audioamplifier and loudspeaker) and the response of each blade may bemeasured with a laser vibrometer 40 coupled to a spectrum analyzer 42,which can be used to analyze the output of the laser vibrometer 40 todetermine the transfer function. The devices 38, 40, and 42 may beobtained from any commercially available sources as is known in the art.For example, the companies that make the function generator and spectrumanalyzer include Hewlett-Packard, Agilent, and Tektronix. The laservibrometer may be a Polytec or Ometron vibrometer.

All of the devices 38, 40, 42 used in the test setup 32 are showncoupled (directly or indirectly through another device) to a computer44, which may be used to operate the devices as well as to analyzevarious data received from the devices. The computer 44 may also storethe FMM software 46, which can include software to implement any or allof the FMM ID methods. It is understood by one skilled in the art thatthe FMM software module 46 may be stored on an external magnetic,electro-magnetic or optical data storage medium (not shown) such as, forexample, a compact disc, an optical disk, a floppy diskette, etc. Thedata storage medium may then be supplied to the appropriate reader unitin the computer 44 or attached to the computer 44 to read the content ofthe data storage medium and supply the FMM software to the computer 44for execution. Alternatively, the FMM software module 46 may reside inthe computer's internal memory such as, for example, a hard disk drive(HDD) from which it can be executed by the computer's operating system.It is apparent to one skilled in the art that the computer 44 may be anycomputing unit including, for example, a stand-alone or networked IBM-PCcompatible computer, a computing work station, etc.

It is noted here that for the sake of convenience and brevity thefollowing discussion uses the term “FMM ID” to refer to any of the basicas well as the advanced FMM ID methods without specifically identifyingeach one. However, based on the context of the discussion and thediscussion presented hereinbefore, it would not be difficult for oneskilled in the art to comprehend which one of the two FMM ID methods isbeing referred to in the discussion.

To investigate applicability of FMM ID methods to real experimental datafrom actual hardware, the methods were applied to a pair of transoniccompressors whose corresponding test rotors were designated as SN-1 andSN-3. A single blade/disk sector finite element model of the tunedcompressor was provided by Pratt and Whitney. By solving this model withfree boundary conditions at the hub and various cyclic symmetricboundary conditions on the radial boundaries of the disk, a nodaldiameter map of the tuned rotor was generated as illustrated in FIG. 23.The free boundary conditions at the hub represented the boundaryconditions in the experiment: an IBR supported by a soft foam pad and isotherwise unconstrained. In FIG. 23, each of the first two families ofmodes (designated by reference numerals 50 and 52) have isolatedfrequencies. These correspond to first bending and first torsion modes,respectively. Because FMM ID is equally applicable for isolated familiesof modes, both the first bending and first torsion modes were suitablecandidates for system identification analysis.

FIG. 24 illustrates a typical transfer function from compressor SN-1obtained using the test setup 32 shown in FIG. 22. Note that due to thehigh modal density, it was necessary to measure the compressor frequencyresponse with a very high frequency resolution. This process wasrepeated for both compressors over two frequency bands to capture theresponse of both the first bending and first torsion modes. Thecommercially available MODENT modal analysis package was then used tocurve-fit the transfer functions. This resulted in measurements of themistuned first bending and torsion modes of each rotor, along with theirnatural frequencies. Because the blade that was excited was at a lowresponse point in some modes, two or three of the modes in each familywere not measurable. In any event, the measured mistuned modes andnatural frequencies were used to demonstrate the applicability of FMM IDto actual hardware.

3.1 FMM ID Results

The measured modes and frequencies were used to test both forms of theFMM ID method. The basic and advanced FMM ID methods were applied toeach rotor, for both the first bending and torsion families of modes.The tuned frequencies required by basic FMM ID were the same as thosedepicted in FIG. 23. To assess the accuracy of FMM ID, the results werecompared to benchmark data.

3.1.1 Benchmark Measure of Mistuning

To assess the accuracy of the FMM ID method, the results must becompared to benchmark data. However, because the test rotors wereintegrally bladed, their mistuning could not be measured directly.Therefore, an indirect approach was used to obtain the benchmarkmistuning. Pratt and Whitney personnel carefully measured the geometryof each blade on the two rotors and calculated the frequencies that itwould have if it were clamped at its root. Because each blade had aslightly different geometry, it also had slightly different frequencies.Thus, the variations in the blade frequencies caused by geometricvariations were determined. This data was put in a form that could becompared with the values identified by FMM ID. First, the frequencyvariations as a fraction of the mean were calculated so that thedeviation in the blade frequencies could be determined. These in turnwere related to the sector frequency deviations determined by FMM ID.For modes with most of their strain energy in the blade, sectorfrequency deviations can be obtained from blade frequency mistuning bysimple scaling, i.e.Δω_(ψ)=α(Δω_(b))  (50)where α is the fraction of strain energy in the blade for the averagenodal diameter mode.

3.1.2 FMM ID Results for Bending Modes

SN-1 Results

The measured mistuned modes and natural frequencies for the compressorSN-1 were used as input to both versions of FMM ID. In the case of basicFMM ID, the tuned system frequencies of the first bending family fromFIG. 23 were also used as input. FIG. 25 illustrates a comparison ofmistuning from each FMM ID method with benchmark results for a testrotor SN-1. FIG. 25 thus shows the sector frequency deviationsidentified by each FMM ID method along with the benchmark results. BothFMM ID methods were in good agreement with the benchmark. This may implythat the mistuning was predominantly caused by geometric variations andthat the variations were accurately captured by Pratt and Whitney.

To make the comparisons easier, all mistuning in FIG. 25 was plotted asthe variation from a zero mean. However, it is noted that rotor SN-1 hada mean frequency 1.3% higher than that of the tuned finite elementmodel. This DC shift was detected by basic FMM ID as a constant amountof mistuning added to each blade's frequency. However, because theadvanced FMM ID formulation does not incorporate the tuned finiteelement frequencies, it had no way to distinguish between a mean shiftin the mistuning and a corresponding shift in the tuned systemfrequencies. Therefore, in advanced FMM ID, mistuning was defined tohave a zero mean, and then a corresponding set of tuned frequencies wasinferred.

FIG. 26 shows a comparison of tuned system frequencies for the testrotor SN-1 from advanced FMM ID (i.e., identified by advanced FMM ID)and the finite element model (FO) using ANSYS® software. It is seen fromFIG. 26 that the FMM ID frequencies were approximately 17 Hz higher thanthe finite element values. This corresponds to a 1.3% shift in the meanof the tuned system frequencies that compensated for fact that the blademistuning now had a zero mean. To facilitate the comparison of thefinite element and FMM ID results, the mean shift was subtracted andthen the results were then plotted as circles on FIG. 26. After thisadjustment, it is seen that the distribution of the tuned frequenciesdetermined by FMM ID agreed quite well with the values calculated fromthe finite element model. Advanced FMM ID additionally identified thefact that SN-1 had slightly higher average frequencies than the FEMmodel—a fact that could be important in establishing frequency marginsfor the stage.

It is observed from the sector frequency deviations of SN-1 shown inFIG. 25 that the mistuning varied from blade-to-blade in a regularpattern. The decreasing pattern of mistuning and the jump in the patternmay suggest that the mistuning might have been caused by tool wearduring the machining process and that an adjustment in the process wasmade during blade manufacturing.

SN-3 Results

The basic and advanced FMM ID methods were then applied in a similarmanner to rotor SN-3's family of first bending modes. The identifiedmistuning and tuned system frequencies are shown in FIGS. 27 and 28,respectively. For comparison purposes, the mistuning was again plottedwith a zero mean, and a corresponding mean shift was subtracted from thepredicted tuned system frequencies. The predictions for rotor SN-3 fromboth FMM ID methods were also in good agreement with the benchmarkresults. It is noted that in FIG. 27, the blades were numbered so thatblade-1 corresponded to the high frequency sector. A similar numberingscheme (not illustrated here) was also implemented for SN-1 forcomparison.

3.1.3 FMM ID Results for Torsion Modes

In this section, FMM ID's ability to identify mistuning in the firsttorsion modes is examined. For brevity, only the results for advancedFMM ID are presented. Advanced FMM ID was applied to each test rotor'sfamily of torsion modes. FIGS. 29 (a) and (b) show a comparison, forrotors SN-1 and SN-3 respectively, of the mistuning identified by FMM IDwith the values from benchmark results obtained by Pratt & Whitney fromgeometric measurements. The agreement between FMM ID and benchmarkresults is good. In FIG. 29, the blades were numbered in the same orderas in FIG. 27, which represents the numbering for SN-3 but, although notshown, a similar numbering for SN-1 was also employed. Thus, themistuning patterns in the torsion modes looked very similar to thoseobserved for the bending modes, e.g., the blades with the highest andlowest frequencies were the same for both sets of modes. This suggeststhat the mistuning in SN-1 and SN-3 systems might have been caused byrelatively uniform thickness variations in the blades because suchmistuning would affect the frequencies of both types of modes in a verysimilar manner.

In addition to identifying the mistuning in these rotors, advanced FMMID also simultaneously inferred the tuned system frequencies of thesystem's torsion modes, as shown in FIG. 30, which illustrates acomparison of tuned system frequencies from advanced FMM ID and ANSYS®software for torsion modes of rotors SN-1 and SN-3. Thus, FMM ID workedwell on both the torsion and bending modes of the test compressors.

3.2 Forced Response Prediction

The mistuning identified in section 3.1 was used to predict the forcedresponse of the test compressors (SN-1, SN-3) to a traveling waveexcitation. The results were compared with benchmark measurements doneby Pratt & Whitney.

Pratt and Whitney has developed an experimental capability forsimulating traveling wave excitation in stationary rotors. Theirtechnique was applied to SN-1 to measure its first bending family'sresponse to a 3E excitation (third engine order excitation). Theresponse of SN-1 was then predicted using FMM ID methods. To make theprediction, the mistuning and tuned system frequencies identified byadvanced FMM ID (as discussed in section 3.1) were input to the FMMreduced order model discussed hereinabove under part [1]. FMM calculatedthe system's mistuned modes and natural frequencies. Then, modalsummation was used to calculate the response to a 3E excitation. Themodal damping used in the summation was calculated from the half-powerbandwidth of the transfer function peaks.

FIG. 31(a) depicts FMM-based forced response data, whereas FIG. 31(1 b)depicts the experimental forced response data. Thus, the plots in FIG.31 show the comparison of the benchmark forced response results withthat predicted by FMM. For clarity, only the envelope of the bladeresponse is shown in FIGS. 31 (a), (b). Also, the plots in FIG. 31 havebeen normalized so that the maximum response is equal to one. Ingeneral, the two curves in FIG. 31 agree reasonably well. To observe howwell the response of individual blades was predicted, the relativeresponses of the blades at two resonant peaks were compared. The peaksare labeled {circle around (1)} and {circle around (2)} in FIG. 31(a).FIGS. 32(a) and (b) respectively show relative blade amplitudes atforced response resonance for the resonant peaks labeled {circle around(1)} and {circle around (2)} in FIG. 31 (a). The relative amplitude ofeach blade as determined by FMM and experimental methods is plotted forboth resonant peaks in FIG. 32. The agreement between FMM andexperimental predictions was reasonably good. Thus, the FMM based methodmay be used to not only capture the overall shape of the response, butalso to determine the relative amplitudes of the blades at the variousresonances.

3.3 Cause and Implications of Repeated Mistuning Pattern

The mistuning in bladed disks is generally considered to be a randomphenomenon. However, it is seen from the discussion in section 3.1 thatboth test rotors SN-1 and SN-2 had very similar mistuning patterns thatwere far from random. If such repeated mistuning matters are found to becommon among IBRS, it may have broad implications on the predictabilityof these systems.

3.3.1 Cause of Repeated Mistuning

The similarity between the mistuning patterns identified in SN-1 andSN-3 is highly suggestive that the mistuning was caused by a consistentmanufacturing effect. In addition, it was observed that the mistuning inthe torsion modes followed the same trends as in the bending modes.Thus, the dominant form of mistuning may most likely be caused byrelatively uniform blade-to-blade thickness variations. Blade thicknessvariations may be analyzed using geometry measurements of a rotor toextract the thickness of each blade at different points across theairfoil. Then, a calculation may be performed to determine how much eachpoint's thickness deviated from the average values of all correspondingpoints. The results can be expressed as a percentage variation from themean blade thickness. It was found that a 2% change in blade thickness,produced about a 1% change in corresponding sector frequency, which isconsistent with beam theory for a beam of curved cross-section.

It is observed that tool wear may cause blade thickness variations. Forexample, if the blades were machined in descending order from blade 18to blade 1 (e.g., the 18 blades in rotor SN-1), then, due to tool wear,each subsequent blade would be slightly larger than the previous one.This effect would cause the sector frequencies to monotonically increasearound the wheel. Any frequency jump or discontinuity observed (e.g.,the jump at blade 15 in FIG. 25) may be the result of a tool adjustmentmade during the machining process.

3.3.2 Implications of Repeated Mistuning

The repeating mistuning patterns caused by machining effects may allowprediction of the response of a fleet (e.g., of compressors) throughprobabilistic methods. For example, consider an entire fleet of thetransonic compressors, two of which—SN-1 and SN-2—were discussedhereinbefore. If it is incorrectly assumed that the mistuning in theserotors was completely random, then one would estimate that the sectorfrequency deviation of each sector has a mean of zero and a standarddeviation of about 2%. Assuming these variations, FMM was used toperform 10,000 Monte Carlo simulations to represent how a fleet ofengines would respond to a 3E excitation. The data from Monte Carlosimulations was used to compute the cumulative probability function(CPF) of the maximum blade amplitude on each compressor in the fleet.FIG. 33 depicts cumulative probability function plots of peak bladeamplitude for a nominally tuned and nominally mistuned compressor. TheCPF of a fleet of engines with random mistuning had a standard deviationof 2% as shown by the dashed line in FIG. 33. It is observed from FIG.33 that the maximum amplitude varied widely across the fleet, ranging inmagnification from 1.1 to 2.5.

However, the test rotors were in fact nominally mistuned with a smallrandom variation about the nominal pattern. Because the random variationwas much smaller than that considered above, the fleet's response wasmore predictable. To illustrate this point, the nominal mistuningpattern (of the fleet of rotors) was approximated as the mean of thepatterns measured for the two test rotors SN-1 and SN-2. Based on thisapproximated pattern, it was found that the sector frequency deviationsdiffered from the nominal values with a standard deviation of only 0.2%,as shown in FIG. 34, which shows mean and standard deviations of eachsector's mistuning for a nominally mistuned compressor. Making use ofthe fact that the rotors were nominally mistuned, the Monte Carlosimulations were repeated. The CPF of the maximum amplitude on eachrotor was then calculated. The calculated results were plotted as thesolid line on FIG. 33. It is observed that by accounting for nominalmistuning, the range of maximum amplitudes is significantly reduced.Thus, by measuring and making use of nominal mistuning when it occurs, atest engineer may predictably determine the fleet's vibratory responsebehavior from the vibratory response of a specific IBR that is tested ina spin pit, rig test or engine.

[4] MISTUNING EXTRAPOLATION FOR ROTATION

The FMM ID methods presented earlier in part [3] determine the mistuningin a bladed disk while it is stationary. However, once the rotor isspinning, centrifugal forces can alter its effective mistuning. However,an analytical method, discussed below, may be used for approximating theeffect of rotation speed on mistuning.

4.1 Mistuning Extrapolation Theory

Centrifugal effects cause the sector frequency deviations present underrotating conditions to differ from their values when the bladed disk isnot rotating. To approximate the effect of rotational speed onmistuning, a lumped parameter model 54 of a rotating blade, as shown inFIG. 35, may be considered. The pendulum 56 mounted on a torsion spring58 represents the blade, while the circular region 60 of the systemrepresents a rigid disk. Thus, the blade is modeled as a pendulum 56 ofmass “m” and length “1” which is mounted to a rigid disk 60 through atorsional spring “k” 58. The disk 60 has radius “L” and rotates at speed“S”.

It can be shown that the blade's natural frequency in this system isgiven by the expression $\begin{matrix}{{\omega(S)}^{2} = {\frac{k}{{ml}^{2}} + {\frac{L}{l}S^{2}}}} & (51)\end{matrix}$where S is the rotation speed in radians/sec, and the notation ω(S)indicates the natural frequency at speed S. Notice that the quantityk/ml² is the natural frequency of the system at rest. Therefore,equation (51) can be rewritten in the more general formω(S)²=ω(0)² +rS ²  (52)where r is a constant.

Take ω to be a mistuned frequency in the form ω(S)=ω°(S)[1+Δω(S)].Substituting this expression into equation (52) and keeping only thefirst order terms impliesω(S)²≈ω°(S)²+2(Δω(0))ω°(0)²  (53)where ω°(S) is the tuned frequency at speed.

Taking the square root of expression (53) and again keeping only thefirst order terms one obtains an expression for the mistuned frequencyat speed, $\begin{matrix}{{\omega(S)} \approx {{\omega^{\circ}(S)}\left\{ {1 + {\Delta\quad{{\omega(0)}\left\lbrack \frac{{\omega^{\circ}(0)}^{2}}{{\omega^{\circ}(S)}^{2}} \right\rbrack}}} \right\}}} & (54)\end{matrix}$Subtracting and dividing both sides of the expression (54) by ω°(S)yields an approximation for the mistuned frequency ratio at speed, i.e.$\begin{matrix}{{\Delta\quad{\omega(S)}} \approx {\Delta\quad{{\omega(0)}\left\lbrack \frac{{{\omega{^\circ}}(0)}^{2}}{{{\omega{^\circ}}(S)}^{2}} \right\rbrack}}} & (55)\end{matrix}$In the case of system modes in which the strain energy is primarily inthe blades, the tuned system frequencies tend to increase with speed bythe same percentage as the blade alone frequencies. Therefore,expression (55) can also be approximated by noting how a frequency ofthe tuned system changes with speed, e.g., $\begin{matrix}{{{\Delta\omega}(S)} \approx {\Delta\quad{{\omega(0)}\left\lbrack \frac{{\omega_{\psi}^{\circ}(0)}^{2}}{{\omega_{\psi}^{\circ}(S)}^{2}} \right\rbrack}}} & (56)\end{matrix}$where ω°_(ψ) is the average tuned system frequency. Expression (56) maythen be used to adjust the sector frequency deviations measured at restfor use under rotating conditions.

4.2 Numerical Test Cases

This section presents two numerical tests of the mistuning extrapolationtheory. The first example uses finite element analysis of the compressorSN-1 discussed hereinbefore (see, for example, FIG. 22) to assess theaccuracy of expression (56). Then, the second example demonstrates thatthis result may be combined with FMM ID and the FMM forced responsesoftware to predict the response of a rotor at speed.

4.2.1 Compressor SN-1

As mentioned earlier, Pratt & Whitney personnel made carefulmeasurements of each blade's geometry and used this data to constructaccurate finite element models of all 18 airfoils in SN-1. Thus, thesefinite element models captured the small geometric variations from oneblade to the next.

Two of the airfoil models were randomly selected for use in this testcase. For the purpose of this study, the first airfoil represented thetuned blade geometry, and the second represented a mistuned blade. Then,both blades were clamped at their root, and their natural frequencieswere calculated using finite element analysis (FEA) software ANSYS®. Thevalues were obtained for the first three modes corresponding to firstbending, first torsion, and second bending respectively. Thecalculations were then repeated with the addition of rotational velocityloads to simulate centrifugal effects. Through this approach, thenatural frequencies of both blades were obtained at five rotation speedsranging from 0 to 20,000 RPM.

Next, the frequency deviation of the mistuned blade was calculated bysubtracting the tuned frequencies from the mistuned values, and thendividing each result by its corresponding tuned frequency. FIG. 36 showsa comparison of mistuning values analytically extrapolated to speed withan FEA (finite element analysis) benchmark. The results plotted as linesin FIG. 36 represent benchmark values on which to assess the accuracy ofthe analytical mistuning extrapolation method. Using expression (55),the frequency deviations calculated for the stationary rotor wereextrapolated to the same rotational conditions considered in thebenchmark calculation. The extrapolated results are shown as circles onFIG. 36. The agreement between extrapolated results and the resultsusing the FEA benchmark were good for all three modes. Thus, expression(55) may be used to analytically extrapolate blade frequency deviationsto rotating conditions. It is noted that expression (55) was used hererather than expression (56) because the calculated natural frequenciesrepresented an isolated blade and not a blade/disk sector. However, forthe cases where FMM is applicable, a blade-alone frequency differs froman average sector frequency by a multiplicative constant. Thus,expression (56) may also be suitable for mistuning extrapolation.

4.2.2 Response Prediction at Speed

This section uses a numerical test case that shows how FMM ID,expression (56), and the FMM forced response software can be combined topredict the response of a bladed disk under rotating conditions.

The geometrically mistuned rotor illustrated in FIG. 14 had a 6^(th)engine order crossing with the first bending modes at a rotational speedof 20,000 RPM. However, to create a more severe test case, it is assumedthat the crossing occurred at 40,000 RPM.

To use FMM to predict the rotor's forced response at this speed (40,000RPM), the FMM prediction software must be provided with the bladeddisk's tuned system frequencies and the sector frequency deviations thatare present at 40,000 RPM. As part of the discussion in section(2.1.3.1) above, these two sets of parameters were determined forat-rest condition using ANSYS and basic FMM ID respectively. However,because both of these properties change with rotation speed, they mustfirst be adjusted to reflect their values at 40,000 RPM.

To adjust the tuned system frequencies for higher rotational speed,tuned system frequencies were recalculated in ANSYS® software using thecentrifugal load option to simulate rotational effects. FIG. 37illustrates the effect of centrifugal stiffening on tuned systemfrequencies. As shown in FIG. 37, the centrifugal stiffening caused thetuned system frequencies to increase by about 30%. Then, the change inthe five nodal diameter, tuned system frequency and expression (56) wereused to analytically extrapolate the sector frequency deviations to40,000 RPM. The adjusted mistuning, along with the original mistuningvalues identified at rest, are plotted in FIG. 38, which illustrates theeffect of centrifugal stiffening on mistuning. It is seen from FIG. 38that the centrifugal loading reduces the mistuning ratios by 40%.

The adjusted parameters were then used with the FMM forced responsesoftware to calculate the rotor's response to a 6E excitation using themethod described hereinabove in parts [1] and [2]. As a benchmark, theforced response was also calculated directly in ANSYS® software using afull 360° mistuned finite element model. Tracking plots of the FMM andANSYS® software results are shown in FIG. 39, which depicts frequencyresponse of blades to a six engine order excitation at 40,000 RPMrotational speed. For clarity, the response of only three blades isshown in FIG. 39: the high responding blade, the median respondingblade, and the low responding blade. It is observed from FIG. 39 thateach blade's peak amplitude and the shape of its overall response aspredicted by FMM agree well with the benchmark results. Thus, FMM ID,the mistuning extrapolation equation, and FMM may be combined toidentify the mistuning of a rotor at rest, and use the mistuning topredict the system's forced response under rotating conditions.

[5] SYSTEM IDENTIFICATION FROM TRAVELING WAVE RESPONSE MEASUREMENTS

Traditionally, mistuning in rotors with attachable blades is measured bymounting each blade in a broach block and measuring its naturalfrequency. The difference of each blade's frequency from the mean valueis then taken as a measure of its mistuning. However, this method cannotbe applied to integrally bladed rotors (IBRs) whose blades cannot beremoved for individual testing. In contrast, FMM ID systemidentification techniques rely on measurements of the bladed disk systemas a whole, and are thus well suited to IBRs.

FMM ID may also be used for determining the mistuning in conventionalbladed disks. Even when applied to bladed disks with conventionallyattached blades, the traditional broach block method of mistuningidentification is limited. In particular, it does not take into accountthe fact that the mistuning measured in the broach block may besignificantly different from the mistuning that occurs when the bladesare mounted on the disk. This variation can arise because each blade'sfrequency is dependent on the contact conditions at the attachment. Inthe engine, the attachment is loaded by centrifugal force from the bladewhich provides a different contact condition than the clamping actionused in broach block tests. This difference is accentuated inmulti-tooth attachments because different teeth may come in contactdepending on how the attachment load is applied. In addition, thecontact in multi-tooth attachments may be sensitive to manufacturingvariations and, consequently, vary from one location to the next on thedisk. The discussion given below addresses these issues by devising amethod of system identification that can be used to directly determinemistuning while the stage is rotating, and can also identify mistuningfrom the response of the entire system because the blades are inherentlycoupled under rotating conditions. The method discussed below providesan approach for extracting the mistuned modes and natural frequencies ofthe bladed disk under rotating conditions from its response to naturallyoccurring, engine order excitations. The method is a coordinatetransformation that makes traveling wave response data compatible withthe existing, proven modal analysis algorithms. Once the mistuned modesand natural frequencies are known, they can be used as input to FMM IDmethods.

5.1 Theory

Both of the FMM ID mistuning identification methods require the mistunedmodes and natural frequencies of the bladed disk as input. Understationary conditions, they can be determined by measuring the transferfunctions of the system and using standard modal analysis procedures.One way of measuring the transfer functions is to excite a single point(e.g., on a blade) with a known excitation and measure the frequencyresponse of all of the other points that define the system. However,when the bladed disk is subjected to an engine order excitation all ofthe blades are simultaneously excited and it may not be clear how theresulting vibratory response can be related to the transfer functionstypically used for modal identification. As discussed below, if theblade frequency response data is transformed in a particular manner thenthe traveling wave excitation constitutes a point excitation in thetransform space and that standard modal analysis techniques can then beused to extract the transformed modes. Once the transformed modes aredetermined, the physical modes of the system can be calculated from aninverse transformation.

5.1.1 Traditional Modal Analysis

Standard modal analysis techniques are based on measurements of astructure's frequency response functions (FRFs). These frequencyresponse functions are then assembled as a frequency dependant matrix,H(ω), in which the element H_(i,j)(ω) corresponds to the response ofpoint i to the excitation of point j as discussed, for example, inEwins, D. J., 2000, Modal Testing: Theory, Practice, and Application,Research Studies Press Ltd., Badlock, UK, Chapter 1. Traditional modalanalysis methods require that one row or column of this frequencyresponse matrix be measured. In the test cases discussed hereinbelow themistuned modes correspond to a single isolated family of modes. Forexample, the lower frequency modes such as first bending and firsttorsion families often have frequencies that are relatively isolated.When this is the case the “modes” of interest may be defined in terms ofhow the blade displacements vary from one blade to the next around thewheel and can be characterized by the response of one point per blade.Thus, the standard modal analysis experiment may be performed in one oftwo ways when measuring the mistuned modes of a bladed disk. First, thestructure's frequency response may be measured at one point on eachblade, while it is excited at only one blade. This would result in themeasurement of a single column of H(ω). Alternatively, a row of H(ω) maybe obtained by measuring the structure's response at only one blade andexciting the system at each blade in turn. In either of these acceptabletest configurations, the structure is excited at only one point at atime. However, in a traveling wave excitation, all blades are excitedsimultaneously. Thus, the response of systems subjected to suchmulti-point excitations cannot be directly analyzed by standard SISO(single input, single output) modal analysis methods.

5.1.2 General Multi-Point Excitation Analysis

As discussed above, a traveling wave excitation is not directlycompatible with standard SISO modal analysis methods. Further, atraveling wave excites each measurement point with the same frequency atany given time. The method discussed below may be applicable to anymulti-input system, in which the frequency profile is consistent fromone excitation point to the next; however, the amplitude and phase ofthe excitation sources may freely vary spatially. It is noted thatsuitable excitation forms include traveling waves, acoustic pressurefields, and even shakers when appropriately driven.

In typical applications, the ij element of the frequency response matrixH(ω) corresponds to the response of point i to the excitation of pointj. However, to analyze frequency response data from a multi-pointexcitation, H_(i,j)(ω) may be viewed in a more general fashion. Thus, ina more general sense, the i,j element describes the response of thei^(th) coordinate to an excitation at j^(th) coordinate. Although thesecoordinates are typically taken to be the displacement at an individualmeasurement point, this need not be the case.

The structure's excitation and response can instead be transformed intoa different coordinate system. For example, an N degree-of-freedomcoordinate system can be defined by a set of N orthogonal basis vectorswhich span the space. In this representation, each basis vector is acoordinate. Thus, to perform modal analysis on multi-point excitationdata, it may be desirable to select a coordinate system in which theexcitation is described by just one basis vector. Within this newlydefined modal analysis coordinate system, the structure is subjected toonly a single coordinate excitation. Therefore, when the responsemeasurements are expressed in this same domain, they represent a singlecolumn of the FRF matrix, and can be analyzed by standard SISO modalanalysis techniques. The following section describes how this approachmay be applied to traveling wave excitations.

5.1.3 Traveling Wave Modal Analysis

Consider an N-bladed disk subjected to a traveling wave excitation. Itis assumed that the amplitude and phase of each blade's response ismeasured as a function of excitation frequency. In practice, thesemeasurements may be made under rotating conditions with a Non-intrusiveStress Measurement System (NSMS), whereas a laser vibrometer may be usedin a stationary bench test. For simplicity, only consider onemeasurement point per blade is considered.

It is assumed that the blades are excited harmonically by the force{right arrow over (f)}(ω)e^(iωt), where the vector {right arrow over(f)} describes the spatial distribution of the excitation force.Similarly, the response of each measurement point is given byh(ω)e^(iωt). The components of {right arrow over (f)} and {right arrowover (h)} are complex because they contain phase as well as magnitudeinformation. It is this excitation and response data from which modesshapes and natural frequencies may be extracted. However, for this datato be compatible with standard SISO modal analysis methods, it mustpreferably first be transformed to an appropriate modal analysiscoordinate system.

As indicated in the immediately preceding section, an appropriatecoordinate system that would allow this to occur is one in which thespatial distribution of the force, {right arrow over (f)}, is itself abasis vector. For simplicity, only the phase difference that occurs fromone blade to the next is included in the equation (57) below. In thecase of higher frequency applications, it may be necessary to alsoinclude the spatial variation of the force over the airfoil if more thanone family of modes interact. The spatial distribution of a travelingwave excitation has the form: $\begin{matrix}{f_{E} = {F_{\circ}\begin{bmatrix}{\mathbb{e}}^{0} \\{\mathbb{e}}^{{- {{\mathbb{i}}{(\frac{2\quad g}{N})}}}E} \\\vdots \\{\mathbb{e}}^{{- {{\mathbb{i}}{({N - 1})}}}{(\frac{2\quad g}{N})}E}\end{bmatrix}}} & (57)\end{matrix}$where E is the engine order of the excitation. Therefore, a coordinatesystem whose basis vectors are the N possible values of {right arrowover (f)}, corresponding to all N distinct engine order excitations, 0through N−1, may be used as a basis. The basis vectors are complete andorthogonal.

The vectors {right arrow over (f)} and {right arrow over (h)} aretransformed into this modal analysis coordinate system by expressingthem as a sum of the basis vectors. Denoting the basis vectors as theset {{right arrow over (b)}₀, {right arrow over (b)}₁, . . . , {rightarrow over (b)}_(N-1)}, this summation takes the form, $\begin{matrix}{\overset{\rightarrow}{f} = {\sum\limits_{m = 0}^{N - 1}{{\overset{\_}{f}}_{m}{\overset{\rightarrow}{b}}_{m}}}} & \left( {58\quad a} \right) \\{{\overset{\rightarrow}{h}(\omega)} = {\sum\limits_{m = 0}^{N - 1}{{\overset{\_}{h}(\omega)}_{m}{\overset{\rightarrow}{b}}_{m}}}} & \left( {58\quad b} \right)\end{matrix}$where the coefficients f _(m) and h(ω)_(m) describe the value of them^(th) coordinate in the modal analysis domain. To identify the valuesof these coefficients, orthogonality may be used. This is a generalapproach that may be applicable for any orthogonal coordinate system.However, for the case of traveling wave excitations, the coordinatetransformation may be simplified.

Consider the n^(th) element of the vectors in equations (58). Forconvenience, let all vector indices run from 0 to N−1. Thus, theseelements may be expressed as, $\begin{matrix}{f_{n} = {\sum\limits_{m = 0}^{N - 1}{{\overset{\_}{f}}_{m}{\mathbb{e}}^{{- {{\mathbb{i}}{(\frac{2\quad g}{N})}}}m\quad n}}}} & \left( {59\quad a} \right)\end{matrix}$ $\begin{matrix}{{h(\omega)}_{n} = {\sum\limits_{m = 0}^{N - 1}{{\overset{\_}{h}(\omega)}_{m}{\mathbb{e}}^{{- {{\mathbb{i}}{(\frac{2\quad g}{N})}}}m\quad n}}}} & \left( {59\quad b} \right)\end{matrix}$where the exponential term is the n^(th) component of the basis vector{right arrow over (b)}_(m). Equation (59) is the inverse discreteFourier Transform (DFT⁻¹) of f. This relation allows to state thetransformation between physical coordinates and the modal analysisdomain in the simpler form, $\begin{matrix}{\overset{\rightarrow}{f} = {{DFT}^{- 1}\left\{ \overset{\rightarrow}{\overset{\_}{f}} \right\}}} & \left( {60\quad a} \right) \\{\overset{\rightarrow}{h} = {{DFT}^{- 1}\left\{ \overset{\rightarrow}{\overset{\_}{h}} \right\}}} & \left( {60\quad b} \right)\end{matrix}$and conversely, $\begin{matrix}{\overset{\rightarrow}{\overset{\_}{f}} = {{DFT}\left\{ \overset{\rightarrow}{f} \right\}}} & \left( {61\quad a} \right) \\{\overset{\rightarrow}{\overset{\_}{h}} = {{DFT}\left\{ \overset{\rightarrow}{h} \right\}}} & \left( {61\quad b} \right)\end{matrix}$where DFT is the discrete Fourier Transform of the vector.

By applying equation (61), the force and response vectors may betransformed to the modal analysis coordinate system. Due to the presentselection of basis vectors, the resulting vector {right arrow over (f)}will contain only one nonzero term that corresponds to the engine orderof the excitation, i.e., a 5E excitation (fifth engine order excitation)will produce a nonzero term in element 5 of {right arrow over (f)}. Thisindicates that within the modal analysis domain, only the E^(th)coordinate has been excited. Therefore, {right arrow over (h)}(ω)represents column E of the FRF matrix.

The transformed response data, {right arrow over (h)}(ω), may now beanalyzed using standard SISO modal analysis algorithms. The resultingmodes will also be in the modal analysis coordinate system, and must beconverted back to physical coordinates though an inverse discreteFourier transform given, for example, in equation (60). These identified(mistuned) modes and natural frequencies may in turn be used as inputsto FMM ID methods to determine the mistuning of a bladed disk from itsresponse to an engine order excitation.

There are two further details of this method. First, for the purpose ofconvenience of notation, the indices of all matrices and vectors arenumbered from 0 to N−1. However, most modal analysis software packagesuse a numbering convention that starts at 1. Therefore, an E^(th)coordinate excitation in the present notation corresponds to an(E+1)^(th) coordinate excitation in the standard convention. This mustbe taken into account when specifying the “excitation point” in themodal analysis software. Second, the coordinate transformation describedherein is based on a set of complex basis vectors. Because the modes areextracted in the modal analysis domain they will be highly complex, evenfor lightly damped systems. Thus it may be necessary to use a modalanalysis software package that can properly handle highly complex modeshapes. In one embodiment, the MODENT Suite by ICATS was used.Information about MODENT may be obtained from Imregun, M., et al., 2002,MODENT 2002, ICATS, London, UK, http://www.icats.co.uk.

5.2 Experimental Test Cases

This section presents two experimental test cases of the traveling wavesystem identification technique. In the first example, an integrallybladed fan (IBR) was excited with a traveling wave while it was in astationary configuration. Because the IBR was stationary, it was easierto make very accurate response measurements using a laser vibrometer.Thus, this example may serve as a benchmark test of the traveling waveidentification theory. Then, in the second example, the method'seffectiveness on a rotor that is excited in a spin pit under rotatingconditions is explored. The amplitude and phase of the response weremeasured using an NSMS system; NSMS is a-non-contacting measurementmethod which is commonly used in the gas turbine industry for rotatingtests. The NSMS technology may be used with the traveling wave systemidentification technique to determine the IBR's mistuning from itsengine order response.

5.2.1 Stationary Benchmark

An integrally bladed fan was tested using the traveling wave excitationsystem at Wright Patterson Air Force Base's Turbine Engine FatigueFacility as discussed in Jones K. W., and Cross, C. J., 2003, “TravelingWave Excitation System for Bladed Disks,” Journal of Propulsion andPower, 19(1), pp. 135-141. Because the facility's test system used anarray of phased electromagnets to generate a traveling wave excitation,the bladed disk remained stationary during the test. The experiment wasperformed with the fan placed on a rubber mat to approximate a freeboundary condition. First, the IBR was intentionally mistuned by fixinga different mass to the leading edge tip of each blade with wax. Themasses ranged between 0 and 7 g, and were selected randomly. Then, toobtain a benchmark measure of the mistuned fan's mode shapes, a standardSISO modal analysis test was performed. Specifically, a singleelectromagnet was used to excite one blade over the frequency range ofthe first bending modes while the response was measured at all sixteen(16) blades with a Scanning Laser Doppler Vibrometer (SLDV). The modeswere then extracted from the measured FRFs using the commerciallyavailable MODENT modal analysis package.

Next, to validate the traveling wave modal analysis method, the fan wasexcited using a 5^(th) engine order traveling wave excitation. Again,the response of each blade was measured using the SLDV. The bladeresponses to the traveling wave excitation were transformed usingequation (61) and then analyzed with MODENT to extract the transformedmodes. Because MODENT numbers its coordinates starting at 1 (0E), a 5Eexcitation corresponds to the excitation of coordinate 6. Therefore, inthe mode extraction process, it was specified that the excitation wasapplied at the 6^(th) coordinate. Lastly, equation (60) was used totransform the resulting modes back to physical coordinates.

The modes measured through the traveling wave test were then comparedwith those from the benchmark analysis. FIGS. 40(a), (b) and (c) show acomparison of the representative mode shape extracted from the travelingwave response data with benchmark results. FIG. 40 thus shows severalrepresentative sets of mode shape comparisons that range from nearlytuned-looking modes (e.g., FIG. 40(a)) to modes that are very localized(e.g., FIG. 40(c)). In all cases in FIG. 40, the modes from thetraveling wave and SISO benchmark methods agreed quite well. Inaddition, the natural frequencies were also accurately identified as canbe seen from FIG. 41, which depicts comparison of the naturalfrequencies extracted from the traveling wave response data with thebenchmark results. Thus, the traveling wave modal analysis method may beused to determine the modes and natural frequencies of a bladed diskbased on its response to a traveling wave excitation.

It is discussed below that the resulting modes and natural frequenciescan be used with FMM ID methods to identify the mistuning in the bladeddisk. Because most of the mistuning in the stationary benchmark fan wascaused by the attached masses, to a large extent the mistuning wasknown. Therefore, these mass values may be used as a benchmark withwhich to assess the accuracy of the FMM ID results.

Because the mass values are to be used as a benchmark, the mistuningcaused by the masses must be isolated from the inherent mistuning in thefan. Therefore, a standard SISO modal analysis was first performed onthe rotor fan with the masses removed, and the resulting modal data wasused as input to FMM ID (e.g., advanced FMM ID). This resulted in anassessment of the IBR's inherent mistuning, expressed as a percentchange in each sector's frequency.

Next, an FMM ID analysis was performed of the modes and frequenciesextracted from the traveling wave response of the rotor withmass-mistuning. The resulting mistuning represented the total effect ofthe masses and the IBR's inherent mistuning. To isolate the mass effect,the rotor's nominal mistuning was subtracted. Again, the resultingmistuning was expressed as a percent change in each sector's frequency.

To compare these mistuning values with the actual masses placed on theblade tips, each sector frequency change may be first translated intoits corresponding mass. A calibration curve to relate these twoquantities was generated through two independent methods. First, thecalibration was determined through a series of finite element analysesin which known mass elements were placed on the tip of a blade, and thefinite element model was used to directly calculate the effect of themass elements on the corresponding sector's frequency. It is noted thatin this method a single blade disk sector of the tuned bladed disk withcyclic symmetric boundary conditions applied to the disk was used.Further, changing the phase in the cyclic symmetric boundary conditionhad only a slight effect on the results (the results shown in FIG. 42corresponded to a phase constraint of 90 degrees). While this method wassufficient in this case, there are often times when a finite elementmodel is not available. For such cases, a similar calibration curve canbe generated experimentally by varying the mass on a single blade, andrepeating the FMM ID analysis. This experimental method was performed asan independent check of the calibration. Both approaches gave verysimilar results, as can be seen from the plot in FIG. 42, which shows acalibration curve relating the effect of a unit mass on a sector'sfrequency deviation in a stationary benchmark. For the range of massesused in this experiment, it was found that mass and sector frequencychange were linearly related as shown in FIG. 42. The calibration curveof FIG. 42 was then used to translate the identified sector frequencychanges into their corresponding masses.

FIG. 43 shows the comparison between the mass mistuning identifiedthrough traveling wave FMM ID with the values of the actual massesplaced on each blade tip (i.e., the input mistuning values). As can beseen from FIG. 43, the agreement between the mistuning obtained usingthe traveling wave system identification method and the benchmark valuesis quite good. Thus, by combining the traveling wave modal analysismethod with FMM ID, the mistuning in a bladed disk from its travelingwave response can be determined.

5.2.2 Rotating Test Case

In the example in section 5.2.1, the traveling wave modal analysismethod was verified using a stationary benchmark rotor. However, if themethod is to be applicable to conventional bladed disks, it may bedesirable to make response measurements under rotating conditions. Thissecond test case assesses if the measurement techniques commonly used inrotating tests are sufficiently accurate to be used with FMM ID todetermine the mistuning in a bladed disk.

For this second case, another rotor fan was considered. To obtain abenchmark measure of the rotor's mistuning in its first bending modes,an impact hammer and a laser vibrometer were used to perform a SISOmodal analysis test. The resulting modes and natural frequencies werethen used as input to FMM ID to determine the fan's mistuning.

Next, the fan was tested in the spin pit facility at NASA Glenn ResearchCenter. The NASA facility used an array of permanent magnets to generatean eddy current excitation that drove the blades. The blade response wasthen measured with an NSMS system. For this test, the fan was drivenwith a 7E excitation, over a rotational speed range of 1550 to 1850 RPM.The test was performed twice, at two different acceleration rates. TheNSMS signals were then processed to obtain the amplitude and phase ofeach blade as a function of its excitation frequency. FIGS. 44(a) and(b) show tracking plots of blade amplitudes as a function of excitationfrequency for two different acceleration rates. The NSMS system measuredthe amplitude and phase of each blade once per revolution. Thus, thedata taken at the slower acceleration rate (FIG. 44(b)) had a higherfrequency resolution than that obtained from the faster accelerationrate (FIG. 44(a)). However, in both cases, the data was significantlynoisier than the measurements obtained in the previous example (insection 5.2.1) using an SLDV.

Next, the traveling wave system identification method was applied toextract the mode shapes from the response data. First, the measurementswere transformed to the modal analysis domain by using equation (61),and the mode shapes and natural frequencies were extracted with MODENT.The extracted modes were then transformed back to the physical domainthrough equation (60). Finally, the resulting modes and frequencies wereused as input to FMM ID to identify the fan's mistuning.

The mistuning identified from the two spin pit tests was then comparedwith the benchmark values. FIGS. 45(a) and (b) illustrate the comparisonof the mistuning determined through the traveling wave systemidentification method with benchmark values for two differentacceleration rates. In the case of the faster acceleration rate (FIG.45(a)), the trends of the mistuning pattern were identifiable, but themistuning values for all blades were not accurately determinable. Thereduced accuracy may be attributed to difficulty in extracting accuratemode shapes from data with such coarse frequency resolution. However,the frequency resolution of the data measured at a slower accelerationrate (FIG. 45(b)) was three times higher than the case for fasteracceleration. Thus, when FMM ID was applied to this higher resolutiondata set, the agreement between the traveling wave based ID and thebenchmark values was significantly improved, as can be seen from FIG.45(b). Thus, with adequate frequency resolution, NSMS measurements canbe used to determine the mistuning of a bladed disk under rotatingconditions.

Thus, NSMS measurements (from traveling wave excitation) may be used toelicit system mode shapes (blade number vs. displacement) and naturalfrequencies. The modes and natural frequency data may then be input to,for example, advanced FMM ID to infer frequency mistuning of each bladein a bladed disk and, thus, to predict the disk's forced response.

There are a number of advantages to performing system identificationbased on a bladed disk's response to a traveling wave excitation. First,it allows the use of data taken in a spin pit or stage test to determinea rotor's mistuning. In this way, the identified mistuning may includeall effects present during the test conditions, i.e., centrifugalstiffening, gas loading, mounting conditions, as well as temperatureeffects. The effect of centrifugal loading on conventional bladed disksmay also be analyzed using FMM ID.

Although FMM ID theoretically only needs measurements of one or twomodes, the method's robustness and accuracy may be greatly improved whenmore modes are included. For certain bladed disks, a single travelingwave excitation can be used to measure more modes than would be possiblefrom a single point excitation test. For example, in a highly mistunedrotor that has a large number of localized modes, it may be hard toexcite all of these modes with only one single point excitation test,because the excitation source will likely be at a node of many of themodes. Therefore, to detect all of the mode shapes, the test must berepeated at various excitation points. However, if the system is drivenwith a traveling wave excitation, all localized modes can generally beexcited with just a single engine order excitation. The more localized amode becomes in physical coordinates, the more extended it will be inthe modal analysis coordinate system. Thus in highly mistuned systems,one engine order excitation can often provide more modal informationthan several single point excitations.

The traveling wave system identification method may form the basis of anengine health monitoring system. If a blade develops a crack, itsfrequency will decrease. Thus, by analyzing blade vibration in theengine, the traveling wave system identification method could detect acracked blade. A health monitoring system of this form may use sensors,such as NSMS, to measure the blade vibration. The measurements may befiltered to isolate an engine order response, and then analyzed usingthe traveling wave system identification method to measure the rotor'smistuning, which can be compared with previous measurements to identifyif any blade's frequency has changed significantly, thus identifyingpotential cracks. It may be possible to develop a mode extraction methodthat does not require user interaction—i.e., an automated modal analysismethod which is tailored to a specific piece of hardware.

The traveling wave system identification method discussed hereinabovemay be extended to any structure subjected to a multi-point excitationin which the driving frequencies are consistent from one excitationpoint to the next. This allows structures to be tested in a manner thatmore accurately simulates their actual operating conditions.

[6] CONCLUSION

FIG. 46 illustrates an exemplary process flow depicting various bladesector mistuning tools discussed herein. The flow chart in FIG. 46summarizes how the FMM and FMM ID methods discussed hereinbefore may beused to predict bladed disk system mistuning in stationary as well asrotating disks. For simplicity, the FMM discussion presentedhereinbefore addressed mistuning in mode families that are fairlyisolated in frequency (i.e., first two or three families of modes).Modeling mistuning in these modes may be relevant to the problem offlutter as discussed in Srinivasan, A. V., 1997, “Flutter and ResonantVibration Characteristics of Engine Blades,” Journal of Engineering forGas Turbines and Power, 119, 4, pp. 742-775. However, as mentionedbefore, the applicability of various FMM methodologies discussedhereinbefore may not be necessarily limited to an isolated family ofmodes.

Referring to FIG. 46, measurements (block 68) of the mode shapes andnatural frequencies of a mistuned bladed disk (block 70) may be input tovarious FMM ID methods (block 72) to infer the mistuning in eachblade/disk sector. The advanced FMM ID method can also calculate thenatural frequencies that the system would have if it were tuned, i.e.,was perfectly periodic. The detailed discussion of blocks 68, 70, and 72has been provided under parts 1 through 3 hereinabove. Because modeshapes measurements are generally made on stationary systems, theresulting mistuning often-corresponds to a non-rotating bladed disk.However, centrifugal forces that are present while the disk rotates canalter the mistuning. Thus, mistuning extrapolation (block 74) may beperformed to correct the mistuning from a stationary rotor for theeffects of centrifugal stiffening. Mistuning extrapolation has beendiscussed under part-4 hereinabove. The FMM methodology (including FMMID methods) may be coupled with a modal summation algorithm to calculatethe forced response (block 76) of a bladed disk based on the mistuningidentified in the previous steps. The discussion of forced responseanalysis (block 76) has been provided hereinabove at various locationsunder parts 1 through 3. Further, as discussed under part-5 above, themode shapes and natural frequencies of a bladed disk may be extractedfrom its response to a traveling wave excitation (blocks 78, 80). Thus,by combining the traveling wave modal analysis technique (block 80)(which may use NSMS measurements identified at block 78) with the FMM IDsystem identification methods, mistuning in a bladed disk can bedetermined while the disk is under actual operating conditions.

The vibratory response of a turbine engine bladed disk is very sensitiveto mistuning. As a result, mistuning increases its resonant stress andcontributes to high cycle fatigue. The vibratory response of a mistunedbladed disk system may be predicted by the Fundamental Mistuning Model(Fe) because of its identification of parameters—the tuned systemfrequencies and the sector frequency deviations—that control the modeshapes and natural frequencies of a mistuned bladed disk. Neither thegeometry of the system nor the physical cause of the mistuning may beneeded. Thus, FMM requires little or no interaction with finite elementanalysis and is, thus, extremely simple to apply. The simplicity of FMMprovides an approach for making bladed disks less sensitive tomistuning—at least in isolated families of modes. Of the two parametersthat control the mistuned modes of the system, one is the mistuningitself which has a standard deviation that is typically known from pastexperience. The only other parameters that affect the mistuned modes arethe natural frequencies of the tuned system. Consequently, thesensitivity of the system to mistuning can be changed only to the extentthat physical changes in the bladed disk geometry affect thesefrequencies. For example, if the disk were designed to be more flexible,then the frequencies of the tuned system would be spread over a broaderrange, and this may reduce the system's sensitivity to mistuning.

The FMM ID methods use measurements of the mistuned system as a whole toinfer its mistuning. The measurements of the system mode shapes andnatural frequencies can be obtained in laboratory test through standardmodal analysis procedures. The high sensitivity of system modes to smallvariations in mistuning causes measurements of those modes themselves tobe an accurate basis for mistuning identification. Because FMM ID doesnot require individual blade measurements, it is particularly suited tointegrally bladed rotors. The basic FMM ID method requires the naturalfrequencies of the tuned system as an input. The method is useful forcomparing the change in a components mistuning over time, because eachcalculation will be based about a consistent set of tuned frequencies.The advanced FMM ID method, on the other hand, does not require anyanalytical data. The approach is completely experimental and determinesboth the mistuning and the tuned system frequencies of the rotor.

Effects of centrifugal stiffening on mistuning may be identified on astationary IBR using FMM ID and FMM, and extrapolated to engineoperating conditions to predict the system's forced response at speed.Further, in conventional bladed disks, centrifugal forces may causechanges in the contact conditions at the blade/disk attachment tosubstantially alter the system's mistuning. This behavior may not beaccounted for in the mistuning extrapolation method. In that case, themode shapes and natural frequencies of a rotating bladed disk may beextracted from measurements of its forced response (e.g., traveling waveexcitation) and the results may then be combined with FMM ID todetermine the mistuning present at operating conditions.

It is observed that, besides centrifugal loading, other factors may alsobe present in the engine that can affect its mistuned response. Thesemay include: temperature effects, gas bending stresses, how the disk isconstrained in the engine, and how the teeth in the attachment changetheir contact if the blades are conventionally attached to the disk.Except for the constraints on the disk, these additional effects may berelatively unimportant in integrally bladed compressor stages. The diskconstraints can be taken into account by performing the systemidentification (using, for example, an FMM ID method) on the IBR afterthe full rotor is assembled. Thus, the FMM ID methodology presentedherein may be used to predict the vibratory response of actualcompressor stages so as to determine which blades may be instrumented,interpret test data, and relate the vibratory response measured in theCRF to the vibration that will occur in the fleet as a whole.

The traveling wave modal analysis method discussed hereinbefore maydetect the presence of a crack in an engine blade by analyzing bladevibrations because a crack will decrease a blade's frequency ofvibration. This method, thus, may be used with on-board sensors tomeasure blade response during engine accelerations. The measurements maybe filtered to isolate an engine order response, and then analyzed usingthe traveling wave system identification method. The identifiedmistuning may then be compared with previous results to determine if anyblade's frequency has changed significantly, thus identifying potentialcracks. The FMM and FMM ID methods may be applied to regions of highermodal density using Subset of Nominal Modes (SNM) method.

The foregoing describes development of a reduced order model called theFundamental Mistuning Model (FMM) to accurately predict vibratoryresponse of a bladed disk system. FMM may describe the normal modes andnatural frequencies of a mistuned bladed disk using only its tunedsystem frequencies and the frequency mistuning of each blade/disk sector(i.e., the sector frequencies). If the modal damping and the order ofthe engine excitation are known, then FMM can be used to calculate howmuch the vibratory response of the bladed disk will increase because ofmistuning when it is in use. The tuned system frequencies are thefrequencies that each blade-disk and blade would have were theymanufactured exactly the same as the nominal design specified in theengineering drawings. The sector frequencies distinguish blade-diskswith high vibratory response from those with a low response. The FMMidentification methods-basic and advanced FMM ID methods—use the normal(i.e., mistuned) modes and natural frequencies of the mistuned bladeddisk measured in the laboratory to determine sector frequencies as wellas tuned system frequencies. Thus, one use of the FMM methodology is to:identify the mistuning when the bladed disk is at rest, to extrapolatethe mistuning to engine operating conditions, and to predict how muchthe bladed disk will vibrate under the operating (rotating) conditions.

In one embodiment, the normal modes and natural frequencies of themistuned bladed disk are directly determined from the disk's vibratoryresponse to a traveling wave excitation in the engine. These modes andnatural frequency may then be input to the FMM ID methodology to monitorthe sector frequencies when the bladed disk is actually rotating in theengine. The frequency of a disk sector may change if the blade'sgeometry changes because of cracking, erosion, or impact with a foreignobject (e.g., a bird). Thus, field calibration and testing of the blades(e.g., to assess damage from vibrations in the engine) may be performedusing traveling wave analysis and FMM ID methods together.

It is noted that because the FMM model can be generated completely fromexperimental data (e.g., using the advanced FMM ID method), the tunedsystem frequencies from advanced FMM ID may be used to validate thetuned system finite element model used by industry. Further, FMM and FMMID methods are simple, i.e., no finite element mass or stiffnessmatrices are required. Consequently, no special interfaces are requiredfor FMM to be compatible with a finite element model.

While the disclosure has been described in detail and with reference tospecific embodiments thereof, it will be apparent to one skilled in theart that various changes and modifications can be made therein withoutdeparting from the spirit and scope of the embodiments. Thus, it isintended that the present disclosure cover the modifications andvariations of this disclosure provided they come within the scope of theappended claims and their equivalents.

1. A method, comprising: obtaining frequency response data of each bladein a bladed disk system to a traveling wave excitation; transformingdata related to spatial distribution of said traveling wave excitationand said frequency response data; and determining a set of mistunedmodes and natural frequencies of said bladed disk system using dataobtained from said transformation.
 2. The method of claim 1, whereinsaid traveling wave excitation has a spatially-invariant frequencyprofile.
 3. The method of claim 1, wherein said obtaining includesmeasuring amplitude and phase of displacement of each said blade as afunction of said traveling wave excitation.
 4. The method of claim 1,wherein said determining includes converting said data obtained fromsaid transformation into a set of physical coordinates.
 5. The method ofclaim 1, further comprising: calculating mistuning of a blade in saidbladed disk system and nominal frequency of said bladed disk system whentuned by using said set of mistuned modes and natural frequencies. 6.The method of claim 5, further comprising validating a finite elementmodel of said bladed disk system using said nominal frequencies of thetuned bladed disk system.
 7. The method of claim 5, wherein calculatingsaid nominal frequencies includes calculating said nominal frequenciesusing a finite element model of said bladed disk system treating eachblade in said bladed disk system as identical and also using said set ofmistuned modes and natural frequencies.
 8. The method of claim 5,wherein said calculating includes calculating said nominal frequenciesfor an isolated family of modes of said bladed disk system when tuned.9. The method of claim 5, wherein said calculating is performediteratively.
 10. The method of claim 5, wherein said blade includes acorresponding blade-disk sector in said bladed disk system.
 11. Themethod of claim 10, wherein a mean value of mistuning of at least oneblade-disk sector is zero.
 12. The method of claim 11, wherein saidcalculating includes solving: where {tilde over (B)} is a stacked matrixcomposed from the elements of {right arrow over (β)}_(j), which is avector containing weighting factors that describe the j^(th) mistunedmode as a sum of tuned modes;

is a stacked matrix of Ω°Γ, where Ω° is a diagonal matrix of the nominalfrequencies of said bladed disk system when tuned and Γ_(j) is a matrixcomposed from the elements in the vector {right arrow over (γ)}_(j)where${\overset{\rightarrow}{\gamma}}_{j} = {{\Omega{^\circ}}{\overset{\rightarrow}{\beta}}_{j}}$${\begin{bmatrix}\overset{\sim}{B} & {2\left( \overset{\sim}{\Omega{^\circ}\Gamma} \right)} \\0 & \overset{\rightarrow}{c}\end{bmatrix}\begin{bmatrix}\overset{\rightarrow}{\lambda{^\circ}} \\\overset{\rightarrow}{\overset{\_}{\omega}}\end{bmatrix}} = \begin{bmatrix}\overset{\rightarrow}{\overset{\_}{r}} \\0\end{bmatrix}$ c is a row vector whose first element is 1 and whoseremaining elements are zero; {right arrow over (ω)} is a vector ofmistuning parameters; λ° is a vector of the tuned frequencies squared;and {right arrow over (r)} is the vector given by the following${\Delta\quad\omega_{\psi}^{(s)}} = {\sum\limits_{p = 0}^{N - 1}{{\mathbb{e}}^{{- {\mathbb{i}}}\quad s\quad p\quad\frac{2\quad\pi}{N}}{\overset{\rightarrow}{\omega}}_{p}}}$$\begin{bmatrix}{\left( {{\omega_{1}^{2}I} - {\Omega{^\circ}}^{2}} \right){\overset{\rightarrow}{\beta}}_{1}} \\{\left( {{\omega_{1}^{2}I} - {\Omega{^\circ}}^{2}} \right){\overset{\rightarrow}{\beta}}_{2}} \\\vdots \\{\left( {{\omega_{1}^{2}I} - {\Omega{^\circ}}^{2}} \right){\overset{\rightarrow}{\beta}}_{m}}\end{bmatrix}$ where I is the identity matrix, and ω₁ is the naturalfrequency of the 1^(st) mistuned mode.
 13. The method of claim 12,wherein the vector {right arrow over (ω)} is related to a physicalsector mistuning by the equation P= where Δω_(ω) ^((s)) is the sectorfrequency deviation of the s^(th) blade-disk sector; and ω _(p) is ap^(th) mistuning parameter in the vector {right arrow over (ω)}.
 14. Themethod of claim 1, further comprising: obtaining nominal frequencies ofsaid bladed disk system when tuned; and calculating mistuning of a bladein said bladed disk system from said nominal frequencies and saidmistuned modes and natural frequencies.
 15. The method of claim 14,wherein said obtaining includes obtaining said nominal frequencies foran isolated family of modes of said bladed disk system when tuned. 16.The method of claim 14, wherein said obtaining includes obtaining saidnominal frequencies using a finite element analysis.
 17. The method ofclaim 16, wherein said finite element analysis includes finite elementanalysis of a tuned, cyclic symmetric model of a single blade-disksector in said bladed disk system.
 18. The method of claim 14, whereinsaid blade includes a corresponding blade-disk sector in said bladeddisk system.
 19. The method of claim 1, wherein said obtaining includes:rotating said bladed disk system; and exciting said rotating bladed disksystem with pressure fluctuations.
 20. The method of claim 1, whereinsaid blade includes a corresponding blade-disk sector in said bladeddisk system.
 21. The method of claim 1, wherein said transforming isperformed from a physical coordinates domain to a modal analysis domain.22. The method of claim 21, wherein said transforming is performedaccording to the following equations: $\begin{matrix}{\overset{\rightarrow}{\overset{\_}{f}} = {{DFT}\left\{ \overset{\rightarrow}{f} \right\}}} \\{\overset{\rightarrow}{\overset{\_}{h}} = {{DFT}\left\{ \overset{\rightarrow}{h} \right\}}}\end{matrix}$ where {right arrow over (f)} is a vector that describesthe spatial distribution of said traveling wave excitation; {right arrowover (h)} is a vector that describes the frequency response of eachmeasurement point to said traveling wave excitation; {right arrow over(f)} is a vector that is a discrete Fourier Transform of the forcevector {right arrow over (f)}; and {right arrow over (h)} is a vectorthat is a discrete Fourier Transform of the response vector {right arrowover (h)}.
 23. A computer-readable data storage medium containing aprogram code, which, when executed by a processor, causes said processorto perform the following: receive frequency response data of each bladein a bladed disk system to a traveling wave excitation; transform datarelated to spatial distribution of said traveling wave excitation andsaid frequency response data; and determine a set of mistuned modes andnatural frequencies of said bladed disk system using data obtained fromsaid transformation.